% CHANGES TO FASCICLE V4F5 OF THE ART OF COMPUTER PROGRAMMING
%
% Copyright (C) 2019, 2020, 2021 by Donald E. Knuth
% This file may be freely copied provided that no modifications are made.
% All other rights are reserved.
%
% Three levels of changes to the books are distinguished here:
%
% "\bugonpage" introduces the correction of an error;
% "\amendpage" introduces new material for future editions;
% "\improvepage" introduces ameliorations of lesser importance.
%
% (Changes introduced by \improvepage do not appear in the hardcopy listing.)
%
% Also, "\planforpage" introduces some of the author's half-baked intentions.
%

% NOTE: TO PUT THE INDEX ON A SEPARATE PAGE, RUN THIS WITH THE COMMAND LINE
%   tex "\let\indexeject+ \input err4f5"

\newif\ifall % \alltrue means show the trivial items too
\relax % hook

\def\vertical{|}
\def\inref#1 #{\expandafter\def\csname\vertical#1\endcsname}
\catcode`|=\active
\let|\inref \input \jobname.ref
\catcode`|=12

\input taocpmac % use the format for TAOCP, with modifications below

\def\becomes{\ifmmode\ \hbox\fi{\manfnt y}\ } % wiggly arrow indicates a change

\def\bugonpage#1.#2 #3 (#4) {
  \medbreak\defaultpointsize
  \line{\kern-5pt\llap{\manfnt x}% print a black triangle in left margin
   {\bf Page #2}\enspace #3
   \leaders\hrule\hfill\ \eightrm\date#4.}
  \nobreak\smallskip\iftrue\noindent}
\def\bugoverall#1.#2 #3 (#4) {
  \medbreak\defaultpointsize
  \line{\kern-5pt\llap{\manfnt x}% print a black triangle in left margin
   {\bf Important Changes Throughout!}\enspace
   \leaders\hrule\hfill\ \eightrm\date#4.}
  \nobreak\smallskip\iftrue\noindent}
\def\amendpage#1.#2 #3 (#4) {
  \medbreak\defaultpointsize
  \line{\kern-5pt{\bf Page #2}\enspace #3
   \leaders\hrule\hfill\ \eightrm\date#4.}
  \nobreak\smallskip\iftrue\noindent}
\def\improvepage#1.#2 #3 (#4) {\ifall
  \medbreak\ninepoint
  \line{\kern-6pt{\sl Page #2\enspace #3\/}
   \leaders\hrule\hfill\ \eightrm\date#4.}
  \nobreak\smallskip\noindent}
\def\planforpage#1.#2 #3 (#4) {
  \medbreak\defaultpointsize
  \line{\kern-5pt{\bf Page #2}\enspace #3
   \leaders\hbox to 5pt{\hss.\hss}\hfill\ \eightrm\date#4.}
  \nobreak\smallskip\begingroup\let\endchange=\endgroup\sl\noindent}
\let\endchange=\fi
\def\nl{\par\noindent}
\def\nlh{\par\noindent\hangit}
\def\hangit{\hangindent2em}
\def\cutpar{{\parfillskip=0pt\par}}

\def\date#1.#2.#3.{% convert "yy.mm.dd." to "dd Mon 19yy"
  #3
  \ifcase#2\or Jan\or Feb\or Mar\or Apr\or May\or Jun\or
               Jul\or Aug\or Sep\or Oct\or Nov\or Dec\fi
  \ \ifnum #1<97 \hundred#1\else19#1\fi}
\def\hundred{20} % the "century" for dates before '97

\def\ex #1. [#2]{\ninepoint
  \textindent{\bf#1.}[{\it#2\/}]\kern6pt}
\def\EX #1. [#2]{\ninepoint
  \textindent{\llap{\manfnt x}\bf#1.}[{\it#2\/}]\kern6pt}

\def\foottext#1{\medskip
        \hrule height\ruleht width5pc \kern-\ruleht \kern3pt \eightpoint
        \smallskip\textindent{#1}}
\def\volheadline#1{\line{\hss\cleaders\hbox{\raise3pt\hbox{\manfnt\char'36}}\hfill
       \titlefont\ #1\ \cleaders\hbox{\raise3pt\hbox{\manfnt\char'36}}\hfill\hss}}
\def\refin#1 {\let|\inref \input #1.ref \let|\crossref}

\let\defaultpointsize=\tenpoint

%%%%%%%%%%%%%% opening remarks %%%%%%%%%%%%%%%%%%%%%%%%

\def\lhead{INTRODUCTION}
\let\rhead=\lhead
\titlepage
\volheadline{THE ART OF COMPUTER PROGRAMMING}
\bigskip
\volheadline{ERRATA TO VOLUME 4 FASCICLE 5}
\bigskip

\noindent This document is a transcript of the notes that I have been making
in my personal copy of {\sl The Art of Computer Programming}, Volume~4,
Fascicle~5, since it was first printed in 2019.

\ifall Four levels of updates\dash---``errors,'' ``amendments,'' ``plans,''
   and ``improvements''\dash---appear, indicated by four
\else Three levels of updates\dash---``errors,'' ``amendments,'' and
   ``plans''\dash---appear, indicated by three \fi
different typographic conventions:
\begingroup\def\hundred{17}

\bugonpage 0.666 line 1 (76.07.04)
Technical or typographical errors (aka bugs) are the most critical items, so
they are flagged with a `\thinspace{\manfnt x}\thinspace' preceding the page
number. The date on which I first was told about the bug is shown; this is the
effective date on which I paid the finder's fee. The necessary
corrections are indicated in
a straightforward way. If,~for example, the book says
`$n$' where it should have said `$n+1$', the change is shown thus:
\smallskip
$n$ \becomes $n+1$
\endchange

\amendpage 0.666 line 2 (89.07.14)
Amendments to the text appear in the same format as bugs, but without
the~`\thinspace{\manfnt x}\thinspace'. These are things I wish I had known
about or thought of when I wrote the original text, so I added them later.
The date is the date I drafted the new text.
\endchange

\def\hundred{19}

\planforpage 0.666 line 3 (17.11.20)
Plans for the future represent a third kind of item. In such notes I~sketched
my intentions about things that I wasn't ready to flesh out further when
I~wrote them down. You can identify these items because they're written in
slanted type, and preceded by a bunch of dots
`\hbox to 6em{\leaders\hbox to 5pt{\hss.\hss}\hfill}' leading to the date on
which I recorded the plan in my files.
\endchange

\improvepage 0.666 line 4 (38.01.10)
The fourth and final category\dash---indicated by page and line number in
smaller, slanted type\dash---consists of minor corrections or improvements
that most readers don't want to know about, because they are so trivial.
You wouldn't even
be seeing these items if you hadn't specifically chosen to print the complete
errata list in all its gory details.
Are you sure you wanted to do that?
\endchange

\endgroup
\ifall\else\medskip\ninepoint
My personal file of updates also includes a fourth category of items, not
shown in this list. They are miscellaneous minor corrections or improvements
that most readers don't want to know about, because they are so trivial.
If you really want to see all of the gory details,
you can download the full list from Internet webpage
$$\.{http://www-cs-faculty.stanford.edu/\char`\~knuth/taocp.html}$$
by selecting the ``long form'' of the errata.
\fi

\medskip
\tenpoint
\beginconstruction
The ultimate, glorious, future editions of Volumes 1--5 are works in progress.
Please let me know of any improvements that you think I ought to make.
Send your comments either by snail mail to D.~E. Knuth, Computer
Science, Gates Building 4B, Stanford University, Stanford CA~94305-9045,
or by email to
{\tt taocp{\char`\@}cs.stanford.edu}. (Use email for book suggestions
only, please\dash---all
other correspondence is returned unread to the sender, or discarded,
because I have no time to
read ordinary email.) Although I'm working full time on
Volume~4 these days, I~will try to reply to all such messages
within a year of receipt. Current news about {\sl The Art of Computer
Programming\/} is posted on
$$\.{http://www-cs-faculty.stanford.edu/\char`~knuth/taocp.html}$$
and updated regularly.
\par\endconstruction
\rightline{\dash---Don Knuth, September 2019}

\bigskip
\bigskip
{\quoteformat
I thought there would be not much corrections.
I honestly wrote what I thought, but was most grievously mistaken.
I find the style incredibly bad, \& most difficult to make clear \& smooth.
.\thinspace.\thinspace. How I could have written so badly is quite inconceivable,
but I suppose it was owing to my whole attention being fixed
on general line of argument, \& not on details.
% "on general": sic
All I can say is that I am very sorry.
\author CHARLES DARWIN, letter to John Murray (14 June 1859)
% The Correspondence of Charles Darwin, vol 7 (Cambridge Univ Press 1991) p303

\bigskip
For the first time in many years, I've pulled out a copy
and read a few chapters to see how much my voice may have
changed over time. I confess to wincing every so often at a
poorly chosen word, a mangled sentence, .\thinspace.\thinspace.
I cannot honestly say, however, that the voice in this book is not mine.
\author BARACK OBAMA, {\sl Dreams From My Father\/} (2004)
% preface to the 2004 edition, page ix
\vfill\eject
}

\def\today{\number\day\space\ifcase\month\or
  January\or February\or March\or April\or May\or June\or
  July\or August\or September\or October\or November\or December\fi
  \space\number\year}

%%%%%%%%%%%%%%% CHANGES FOR VOLUME 4 FASCICLE 6 %%%%%%%%%%%%%%%%%%%%%%%%%%%

\def\lhead{CHANGES TO V4F5: MATH PRELIMINARIES, BACKTRACKING ET AL}
\def\rhead{CHANGES TO V4F5: MATH PRELIMINARIES, BACKTRACKING ET AL}
\titlepage
\volheadline{MATH PRELIMINARIES REDUX}   % the fascicle title
\volheadline{INTRO TO BACKTRACKING}      % the fascicle title
\volheadline{DANCING LINKS}              % the fascicle title
\bigskip
\rightline{Copyright \copyright\ 2019, 2020, 2021
Addison\with Wesley}
\rightline{Last updated \today}
\bigskip
%\rightline{\sl Most of these corrections have already been made in
%                  recent printings.}
%\medskip
%\noindent Important note: The changes below include nearly every difference
%between the paperback fascicle and the first printing of Volume 4A,
%published in January 2011. All subsequent changes can be found in
%the errata list for that volume.

\smallskip
\let\defaultpointsize=\tenpoint

\amendpage 4f5.ii line $-4$ (20.02.10)
\ninepoint\tt
 {\tt www.pearsoned.com/permissions/} \becomes\nl
 {\tt www.pearson.com/permissions/}
\endchange

\amendpage 4f5.iv line 19 (20.02.09)
will solve the \becomes will solve many 
\endchange

\amendpage 4f5.viii new contents item (21.09.18)
\ninepoint
\def\0 #1 |#2|{\line{{#1}\leaders\hbox to
1em{\hfil.\hfil}\hfil\hbox to 1.7em{\hfil#2}}}%
\0 {\tenssbx Breaking News} |384|
\endchange

\improvepage 4f5.4 line $-7$ (21.07.26)
with probability~$q$ \becomes with probability~$q=1-p$
\endchange

\improvepage 4f5.13 in exercise {5(b)} (21.02.08)
\ninepoint Suppose $A$ and $B$ \dots\ is odd. \becomes
Suppose the random variable~$A$ takes the values $(m_0,m_2,m_4,\ldots{})$
with probabilities $(q_0,q_2,q_4,\ldots{})$; otherwise $A=0$. Independently,
the random variable~$B$ takes the values $(m_1,m_3,m_5,\ldots{})$
with probabilities $(q_1,q_3,q_5,\ldots{})$; otherwise $B=0$.
\endchange

\amendpage 4f5.14 new rating for exercise 24 (20.10.27)
\ninepoint [{\it\HM27\/}] \becomes [{\it\HM28\/}]
\endchange

\improvepage 4f5.18 line 3 of exercise 63 (20.08.16)
\ninepoint those probabilities \becomes those nine probabilities
\endchange

\bugonpage 4f5.23 line 6 of exercise 112 (20.02.26)
\ninepoint
$\min_{a\in\pi}a\pi=\infty$ \becomes
$\min_{a\in A}a\pi=\infty$
\endchange

\amendpage 4f5.24 line 4 of exercise 121 (21.05.31)
{\it divergence of\/ $Y\!$ from~$X\?$}, \becomes
{\it divergence of\/ $X\!$ from~$Y\!$},
\endchange

\bugonpage 4f5.26 bottom line (21.05.06)
$p_{k-1}\le p_k\ge p_{k-1}$ \becomes
$p_{k-1}\le p_k\ge p_{k+1}$
\endchange

\amendpage 4f5.27 new exercise (21.09.18)
\ninepoint\ex135. [\HM30] ({\it Baxter permutations.})
Let $p_1\ldots p_n$ be a permutation of $\{1,\ldots,n\}$, and let
$q_1\ldots q_n$ be its inverse. These permutations are called Baxter
permutations if and only if
there are no indices $k$ and $l$ with $0<k,l<n$ such that
either ($q_{k+1}<l$ and $q_k>l$ and $p_l<k$ and $p_{l+1}>k$) or
       ($q_k<l$ and $q_{k+1}>l$ and $p_l>k$ and $p_{l+1}<k$).

What's a good way to count the number $b_n$ of $n$-element
Baxter permutations?\tighten
\endchange

\amendpage 4f5.27 new exercise (21.09.26)
\ninepoint
\ex136. [\HM20] Let $f(x)=x\ln x$ be the fundamental convex function
that underlies formulas for entropy. Prove or disprove: If $0\le x\le y\le 1$
then $\vert f(y)-f(x)\vert\le\vert f(y-x)\vert$.
\endchange

\improvepage 4f5.30 line 2 (20.11.14)
level 5 \becomes level 5 below the root.
\endchange

\improvepage 4f5.35 lines $-19$ and $-17$ (20.05.15)
line $-19$: four-letter words \becomes four-letter codewords\nl
line $-17$: from words of the set \becomes from strings of the set
\endchange

\improvepage 4f5.36 lines 2 and 3 (20.05.15)
line 2: Such a word \becomes Such a string\nl
line 3: {\it aperiodic\/} words \becomes {\it aperiodic\/} strings
\endchange

\improvepage 4f5.37 line 15 (20.11.14)
no commafree code \becomes no commafree code of length $(m^4-m^2)/4$
\endchange

\amendpage 4f5.39 new copy for two lines after {\eq(23)} (20.08.16)
\noindent
(This data-structuring technique has been called a {\it sparse-set
representation\/}; see P.~Briggs and L.~Torczon, {\sl ACM Letters
Prog.\ Lang.\ and Syst.\ \bf2} (1993), 59--69.)\tighten
\endchange

\improvepage 4f5.39 line 10 (20.08.16)
codes is not \becomes codes of maximum length is not
\endchange

\bugonpage 4f5.40 line 5 after the table (20.05.30)
$\.{CLOFF}+m^4+cl(\alpha)$. \becomes
$\.{S3OFF}+m^4+s_3(\alpha)$,
$\.{CLOFF}+m^4+4cl(\alpha)$.
\endchange

\bugonpage 4f5.43 line 2 (20.05.30)
we must backtrack \becomes we mustn't proceed
\endchange

\bugonpage 4f5.44 line 2 of step C6 (20.05.30)
$f\gets f-1$ \becomes $f\gets f+1$
\endchange

\bugonpage 4f5.47 line 6 (20.06.05)
$y_1y_2\ldots{}$ \becomes $y_0y_1\ldots{}$
\endchange

\improvepage 4f5.48 line 19 (20.11.14)
shaky grounds \becomes shaky ground
\endchange

\amendpage 4f5.48 line 2 before {\eq(34)} (20.01.19)
P.~Diaconis and S.~Chatterjee \becomes
S.~Chatterjee and P.~Diaconis
\endchange

\amendpage 4f5.48 and 49 (20.01.19)
\eightpoint\noindent
[I'm now using the notation $\widehat\chi_N$ instead of $Q_N$,
for eventual agreement with Section 7.2.2.9]
\endchange

\amendpage 4f5.51 line $-16$ (21.10.27)
a student named \becomes a telegraph engineer named
\endchange

\amendpage 4f5.53 replacement for exercise 15 (21.10.07)
\ninepoint
\ex15. [\HM42] (M. Simkin, 2021.) Show that
$Q(n)\approx \sigma^nn!$ as $n\to\infty$, where $\sigma\approx0.388$.\tighten
% E^(1-1.944)=0.3891, E^(1-1.9449)=0.3887
\endchange

\amendpage 4f5.54 new rating for exercise 19 (20.11.14)
\ninepoint [{\it M19\/}] \becomes [{\it M10\/}]
\endchange

\bugonpage 4f5.55 line 9 of exercise 37 (20.12.22)
\ninepoint
$x_k>x_{k+1}\le x_{k+2}$ \becomes
$x_k\ge x_{k+1}<x_{k+2}$
\endchange

\amendpage 4f5.56 new rating for exercise 46 (20.08.17)
\ninepoint [{\it M30\/}] \becomes [{\it M35\/}]
\endchange

\amendpage 4f5.58 new rating for exercise 69 (19.12.05)
\ninepoint [{\it40\/}] \becomes [{\it41\/}]
\endchange

\bugonpage 4f5.59 in {9(D)} of Table 666 (20.10.24)
\ninepoint $[61\dts67]$ \becomes $[59\dts67]$
\endchange

\amendpage 4f5.60 in lines 4 and 6 of exercise 77 (21.06.01)
\ninepoint $P_3^\to\sqprod P_3^\to$ \becomes
$P\orie_{\!3}\,\sqprod P\orie_{\!3}$
\endchange

\amendpage 4f5.64 line 2 before {\eq(5)} (20.08.17)
find a subset \becomes find all subsets
\endchange

\amendpage 4f5.65, replacement for the third bullet point in {\eq(9)} (20.08.17)
\noindent$$\def\\{$\bullet$\enspace}
\vcenter{
\hbox{\\For each just-deleted option $O$ that involves $i$,
         one at a time, do this:}
\hbox{\indent for each item $j\ne i$ in $O$, cover item $j$; then}
\hbox{\indent recursively append $O$ to each solution of the residual problem.}}
$$
\endchange

\improvepage 4f5.66 just before Table 1 (20.01.01)
are unused. \becomes
are unused and can contain anything.
\endchange

\improvepage 4f5.66 first line of cover$(i)$ (20.01.01)
(Here $p$, $l$, and $r$ are local variables.) \becomes
(Undeclared variables like $p,l,r$ are local.)
\endchange

\improvepage 4f5.71 line 6 (19.11.29)
amendment of \becomes amendment to
\endchange

\improvepage 4f5.78 line 6 before {\eq(37)} (20.11.21)
consecutive letters. \becomes consecutive letters from O to Z.
\endchange

\bugonpage 4f5.84 line 24 (21.06.22)
`$@$\.{4}\enspace\.{000}\enspace\.{011}\enspace\.{111}\enspace
\.{121}\enspace\.{4-}$@$' \becomes
`$@$\.{4}\enspace\.{001}\enspace\.{011}\enspace\.{111}\enspace
\.{121}\enspace\.{4-}$@$'
\endchange

\amendpage 4f5.84 line $-2$ (19.12.21)
Exercises 325--345 \becomes Exercises 324--350
\endchange

\amendpage 4f5.87 line 4 after {\eq(49)} (20.03.27)
never examined. \becomes never examined, except in printouts (see
the answer to exercise 12).
\endchange

\amendpage 4f5.87 line 7 of Table 2 (21.09.19)
\eightpoint
\.{LEN($x$)} \becomes \.{LEN($x$)},\thinspace\.{TOP($x$)}
\endchange

\amendpage 4f5.87 new line for nodes $x=0, 1, 2, 3, 4, 5, 6$ in Table 2 (20.03.27)
\eightpoint
{\def\\{\hbox{---}}\tabskip=6pt plus 1fil
\halign to \hsize{\hfil#:\hfil&&\hfil$#$\cr
\.{COLOR($x$)}&\\&\\&\\&\\&\\&\\& 0\cr
}}
\endchange

\amendpage 4f5.88 in first line of {\eq(55)} (20.03.27)
$q\gets\.{DLINK($i$)}$. \becomes
$\.{COLOR($i$)}\gets c$, $q\gets\.{DLINK($i$)}$.
\endchange

\amendpage 4f5.88 third line of step C5 (20.07.15)
This covers \becomes This commits
\endchange

\amendpage 4f5.88 third line of step C6 (20.07.15)
This uncovers \becomes This uncommits
\endchange

\amendpage 4f5.96 line 1 (20.07.15)
partially covers \becomes commits
\endchange

\amendpage 4f5.96 line 4 of step M7 (20.07.15)
uncovers \becomes uncovers or uncommits
\endchange

\bugonpage 4f5.102 line 6 (21.07.10)
evenually \becomes eventually
\endchange

\bugonpage 4f5.107 line 19 (20.03.07)
`$a$~$g$' and `$a$~$f$'. Further simplifications now arise. \becomes
`$a$~$g$' and `$a$~$f$'.
\endchange

\amendpage 4f5.110 line 1 after {\eq(108)} (21.12.23)
both $\Sigma_1$ and $\Sigma_2$. \becomes
both $\Sigma_1$ and $\Sigma_2$!
\endchange

\improvepage 4f5.122 last two lines of exercise 2 (20.10.24)
\ninepoint say $(a_1,\ldots,a_k)$, \dots $(a_1,\ldots,a_k)$. \becomes\nl
say $a_1$, \dots,~$a_k$,
then undelete them in exactly the same order $a_1$, \dots, $a_k$.
\endchange

\bugonpage 4f5.126 in exercise 49 (21.08.09)
[the bipartite graph should have an edge between $u_7$ and $v_1$]
\endchange

\bugonpage 4f5.127 line 1 of exercise 57 (21.09.10)
\ninepoint has 27 \becomes has at most
\endchange

\amendpage 4f5.130 last line of exercise 76 (21.09.19)
\ninepoint an identity element \becomes the identity element 0
\endchange

\amendpage 4f5.130 an additional sentence in exercise 77 (20.03.02)
(In such a case we say that $G$ is {\it embedded\/} in~$H$.)
\endchange

\amendpage 4f5.135 line 1 of exercise 116 (21.05.25)
\ninepoint
Given a graph on vertices $V$ \becomes
Given a graph $G$ on vertices $V$
\endchange

\amendpage 4f5.135 better illustrations in exercise 116 (20.02.19)
$$\def\\#1{\vcenter{\epsfbox{\figdir/v4b.328#1}}}
M_2\;=\;\\1\,;\qquad
M_3\;=\;\\2\,;\qquad
M_4\;=\;\\3\,;\qquad \ldots\,.$$
\endchange

\bugonpage 4f5.136 in exercise 123 (21.12.23)
\ninepoint line 5: into $m$ parts \becomes into parts\nl
line 6: $+\cdots{n\brace m}m\falling m$ \becomes
           $+\cdots+{n\brace m}m\falling m$
\endchange

\bugonpage 4f5.144 correction to puzzle {(ii)} in exercise 173 (19.12.04)
\ninepoint \dots\ Diagram (ii) shows 8 \dots\ the other 19 can be deduced. \dots
$$\def\epsfsize#1#2{.5#1}\vcenter{\epsfbox{\figdir/v4b.2121}}$$
\endchange

\bugonpage 4f5.144 rewording of exercise 177 (21.11.17)
\ninepoint
\ex177. [M21] Algorithm 7.2.1.5M generates the
$p(n_1,\ldots,n_m)$ partitions of the multiset
$\{n_1\cdot x_1,\ldots,n_m\cdot x_m\}$ into submultisets.
Consider the special cases where $n_1=\cdots=n_s=1$ and
$n_{s+1}=\cdots=n_{s+t}=2$ and $s+t=m$.
\item{a)} Generate those partitions
with Algorithm~M, using the previous two exercises.
\item{b)} Also generate the $q(n_1,\ldots,n_m)$
multipartitions into {\it distinct\/} multisets.
\endchange

\improvepage 4f5.148 in exercise 208 (21.12.19)
\ninepoint
`\.{1a} \.{2a} \.{3a} \.{4a} \.{2b} $V_{\hbox{\seventt 1b}}$' and
`\.{1c} \.{2c} \.{3c} \.{4c} \.{3b} $V_{\hbox{\seventt 1b}}$'
\becomes
`\.{1c} \.{2c} \.{3c} \.{4c} \.{3b} $V_{\hbox{\seventt 1b}}$' and
`\.{1a} \.{2a} \.{3a} \.{4a} \.{2b} $V_{\hbox{\seventt 1b}}$'
\endchange

\amendpage 4f5.150 in exercise 225 (21.12.08)
\ninepoint the $6\times15$ case \becomes the ``one-sided'' $6\times15$ case
\endchange

\bugonpage 4f5.150 in exercise 235 (20.10.18)
\ninepoint costs $\$d(i,j)^4$ \becomes costs $\$4d(i,j)^4$
\endchange

\bugonpage 4f5.155 line 1 of exercise 281 (20.09.02)
\ninepoint $(2n{+}1\times(2n{+}1)$ \becomes $(2n{+}1)\times(2n{+}1)$
\endchange

\improvepage 4f5.155 line 3 of exercise 284 (21.05.25)
\ninepoint (a) $\{I,P,U,V\}$ and (b) $\{F,P,T,U\}$ \becomes
(a) $\{\rm O,P,U,V\}$ and (b) $\{\rm P,R,T,U\}$
\endchange

\bugonpage 4f5.165 the illustrations in exercise {341(a)} and 345 (21.05.25)
[the darker cubies should be much darker; compare with the colored versions
online in {\tt cs.stanford.edu/\char`\~knuth/fasc5c.ps.gz}, page 103]
\endchange

\amendpage 4f5.169 and 170, replacement for exercise 372 (21.09.10)
\ninepoint \EX372. [M35] ({\it Floorplans}.)
If a rectangle decomposition satisfies the tatami condition\dash---``no
four rectangles meet''\dash---it's often called a {\it floorplan}, and its
subrectangles are called {\it rooms}. The line segments that delimit
rooms are called {\it bounds}. Four possibilities arise when room~$r$
is adjacent to bound~$s$: Either
$s\downarrow r$,
$r\rightarrow s$,
$r\downarrow  s$, or
$s\rightarrow r$, meaning respectively that the top, right, bottom, or
left boundary of~$r$ is part of~$s$.

 For example, the floorplans shown
on the next page have 10 rooms $\rm\{A,B,\ldots,J\}$,
$7+6$ bounds $\{h_0,\ldots,h_6,v_0,\ldots,v_5\}$,
and the following adjacencies:
\def\\#1{\hbox{#1}}%
$h_0\downarrow\\A\downarrow h_3\downarrow\\D\downarrow h_5
  \downarrow\\E\downarrow h_6$,
$h_0\downarrow\\B\downarrow h_1\downarrow\\C\downarrow h_3
  \downarrow\\F\downarrow h_6$,
$h_1\downarrow\\G\downarrow h_2\downarrow\\H\downarrow h_4
  \downarrow\\I\downarrow h_6$,
$h_2\downarrow\\J\downarrow h_6$;
$v_0\rightarrow\\A\rightarrow v_1\rightarrow\\B\rightarrow v_5$,
$v_1\rightarrow\\C\rightarrow v_3\rightarrow\\H\rightarrow v_4$,
$v_0\rightarrow\\D\rightarrow v_2\rightarrow\\F\rightarrow v_3
  \rightarrow\\G\rightarrow v_5$,
$v_0\rightarrow\\E\rightarrow v_2$,
$v_3\rightarrow\\I\rightarrow v_4\rightarrow\\J\rightarrow v_5$.

$$\def\\#1{\vcenter{\epsfbox{\figdir/v4b.372#1}}}
\kern-10pt\\0\quad\\1\quad\\2\quad\\3\kern-10pt$$

Two floorplans with the same adjacencies are considered to be equivalent.
Thus, all four of the floorplans above are essentially the same, even though
they look rather different: In particular, room~C needn't overlap room~D;
we require only $\\C\downarrow h_3\downarrow\\D$.\tighten

\item{a)} Let $r\Downarrow r'$ mean that $r=r_0\downarrow s_0\downarrow r_1
\downarrow\cdots\downarrow s_{k-1}\downarrow r_k=r'$ for some $k>0$;
define $r\Rightarrow r'$ similarly. Prove that
$\[r\Downarrow r']+\[r\Rightarrow r']+\[r'\Downarrow r]+\[r'\Rightarrow r]=1$,
when $r\ne r'$.
\em Hint: Every floorplan has unique diagonal and antidiagonal
equivalents, as shown.\tighten

\begingroup\advance\thickmuskip-1mu
\item{b)} A {\it twin tree\/} is a data structure whose nodes~$v$
have four pointer fields, \.{L0($v$)}, \.{R0($v$)},
\.{L1($v$)}, \.{R1($v$)}. It defines
two binary trees $T_0$ and $T_1$ on the nodes, where
$T_\theta$ is rooted at \.{ROOT$\theta$} and has child links
$(\.{L$\theta$},\.{R$\theta$})$.
These two trees satisfy inorder$(T_0)={\rm inorder}(T_1)=
v_1\ldots v_n$;
$\.{R0($v_k$)}=\Lambda$ $\iff$ $\.{R1($v_k$)}\ne\Lambda$,
for $1\le k<n$.\break\indent
For each room~$r$, if $r$'s top left corner is a
$\top$ junction, set $\.{L0($r$)}\gets\Lambda$ and $\.{L1($r$)}\gets r'$,
where $r'$ is the room opposite~$r$ in that corner; otherwise reverse the
roles of \.{L0} and \.{L1}. Similarly, set $\.{R0($r$)}\gets\Lambda$
and $\.{R1($r$)}\gets r'$ if the bottom right corner of~$r$ is a
$\dashv$ junction, or vice versa otherwise. (Use $r'=\Lambda$ at
extreme corners.) Also set $\.{ROOT0}$ and $\.{ROOT1}$ to the
bottom-left and top-right rooms. Show that a twin tree is created,
convenient for representing this floorplan.\tighten
\par\endgroup
\endchange

\amendpage 4f5.170 and 171, in exercise 375 (20.02.10)
\ninepoint smallest possible perimeter \becomes
smallest possible semiperimeter (height plus width)\nl
smallest perimeter \becomes smallest semiperimeter
\endchange

\amendpage 4f5.172, append new sentence to exercise 387 (20.10.27)
\ninepoint (Note that mirror reflection is {\it not\/} a legal symmetry for a
polycube; L-twist $\ne$ R-twist!)
\endchange

\bugonpage 4f5.180 corrected diagram in exercise 433 (21.05.25)
\noindent$$\def\epsfsize#1#2{.7#1}
\vcenter{\epsfbox{\figdir/v4b.3338}}$$
\endchange

\let\defaultpointsize=\ninepoint % get ready for answer pages

\amendpage 4f5.183 last line of answer 2 (21.08.26)
156.] \becomes A.~Komisarski, {\sl AMM \bf128} (2021), 423--434.]
\endchange

\amendpage 4f5.185 in answer 24 (20.10.18)
line 2 becomes
$$B'_{m,n}(x)=\sum_{k=1}^m\Bigl({n\atop k}\Bigr)kx^{k-1}(1-x)^{n-k}
         -\sum_{k=0}^m\Bigl({n\atop k}\Bigr)(n-k)x^k(1-x)^{n-1-k}$$
line 7: It suffices \becomes If $a>0$ it suffices\nl
line 12: increases \becomes is nondecreasing
\endchange

\amendpage 4f5.186 last line of answer 24 (20.10.27)
332.] \becomes
332. See also
S.~Janson, \arXiv:2009.13781 [math.PR] (2020), 12~pages.]
\endchange

\improvepage 4f5.191 new answer 63 (20.08.16)
\ans63. If $\omega$ is the event `$Z_0=a$ and $Z_1=b$', we have
$Z_0(\omega)=a$ and $\E(Z_1\given
Z_0)(\omega)=\break(p_{a1}+2p_{a2})/(p_{a0}+p_{a1}+p_{a2})$. Hence
$p_{01}=p_{02}=p_{20}=p_{21}=0$, and $p_{10}=p_{12}$. Those conditions are
necessary and sufficient for $\E(Z_1\given Z_0)=Z_0$.
\endchange

\bugonpage 4f5.194 in answer 83 (20.08.16)
line 2: $\min(m,N)$ \becomes
$N'_n(x_0,\ldots,x_{n-1})=\[n<m]\cdot N_n(x_0,\ldots,x_{n-1})$\nl
line 3: $S_n$ \becomes $S_N$
\endchange

\amendpage 4f5.205 new answer 135 (21.09.18)
\ansss135, 136, .\thinspace.\thinspace.\thinspace$@$.
 See page 384.
\endchange

\bugonpage 4f5.206 lines 4 and 5 of answer 10 (20.12.22)
$a_l\gets a_{l-1}+t$ \dots\ and \becomes\nl
$a_{l+1}\gets a_l+t$, $b_{l+1}\gets(b_l+t)\rsh1$,
$c_{l+1}\gets((c_l+t)\lsh1)\band\mu$; and
\endchange

\bugonpage 4f5.207 line 2 of answer 12 (20.10.11)
we have also $\sum_{k=1}^n k\equiv{n+1\choose2}$ \becomes
the same sum is also $\equiv{n+1\choose2}$
\endchange

%\amendpage 4f5.207 last line of answer 14 (20.12.22)
%Theorem 2.] \becomes Theorem 2.
%They ask: Does $\lim_{n\to\infty}(\ln Q(n))/(n\ln n)$ exist?
%Is that sequence monotonically increasing, for all sufficiently large~$n$?]
%\endchange

\amendpage 4f5.207 replacement for answer 15 (21.10.07)
%\ans15. See the construction by Z. Luria, \arXiv:1705.05225 [math.CO]
%(2017), 12~pages,~\S2.
\ans15. More precisely, there's a constant $\sigma=e^{1-\alpha}$ such that,
for any fixed $\epsilon$ with $0<\epsilon<\sigma$, $Q(n)/n!$ is q.s.\ between
$((1-\epsilon)@\sigma)^n$ and $((1+\epsilon)@\sigma)^n$.
% the estimate is not great for small n however!
In fact, a subtle analysis [\arXiv:2107.13460 [math.CO] (2021), 51~pages]
shows that the average of all solutions 
approaches a fascinating probability distribution. S.~Boyd, A.~Agrawal,
and P.~Nobel have computed $\alpha$ accurately
[\arXiv:2112.03336 [math.CO] (2021), 14~pages].\tighten
\endchange

\bugonpage 4f5.208 line 3 of answer 24 (20.04.22)
word~$x_3$ \becomes word~$x_l$
\endchange

\bugonpage 4f5.211 line 7 of answer 37 (20.12.22)
$b_{e+1}=a_eb_eb_e$ \becomes
$b_{e+1}=b_ea_eb_e$
\endchange

\bugonpage 4f5.211 line 1 of step O3 (20.12.22)
less $(j-1)=1$ \becomes
less$(j-1)=1$
\endchange

\bugonpage 4f5.211 in step O4 (20.12.22)
If $j<t$, set $b'_{t'}\gets b_j$; \becomes
If $i<t$, set $b'_{t'}\gets b_i$;\nl
$b'_0\gets b_{j-t}$ \becomes $b'_0\gets b_{i-t}$
\endchange

\bugonpage 4f5.211 line 3 of step O5 (19.12.18)
while $b_{k-t}<n$ \becomes while $b_{k-t}<2n$
\endchange

\bugonpage 4f5.212 line 2 of answer 38 (20.12.22)
$z_2=j$ \becomes $z_{k-1}=j$
\endchange

\bugonpage 4f5.212 line 1 of answer 44 (20.05.30)
$r\gets m+1$ \becomes $r\gets5$
\endchange

\bugonpage 4f5.213 line $-5$ of answer 45 (20.05.30)
go to C6 \becomes go to C5
\endchange

\amendpage 4f5.213 replacement for last paragraph of answer 46 (21.02.08)
So the answer is ``No.'' But we can come close: Aaron Windsor used a
{\mc SAT} solver in 2021 to discover a binary commafree code for $n=12$
that contains a representative of every cycle class except [000011001011].
\endchange

\bugonpage 4f5.213 line 8 of answer 47 (20.05.30)
1002, 2201, 2001, \becomes
1002, 1102, 2001,
\endchange

\amendpage 4f5.214 replacement for answer 48 (21.02.08)
\ans48. Algorithm C isn't fast enough to solve this problem.
But Aaron Windsor used a {\mc SAT} solver in 2021 to find such a code
of size $139=(5^4-5^2)/4-11$, and to prove that no such code of size 140 exists.
(He also found, rather quickly, that an optimum ternary commafree
code for $n=6$ contains $(3^6-3^3-3^2+3^1)/6-3=113$ codewords.)
\endchange

\amendpage 4f5.217 line $-3$ of answer 61 (21.09.18)
\#R44. \becomes \#R44, 1--16.
\endchange

\bugonpage 4f5.221 line 3 of answer 73 (19.12.12)
\.{roots} \becomes \.{roofs}
\endchange

\bugonpage 4f5.222 line 1 of step R6 (20.05.12)
set $i\gets i_l$ \becomes set $i\gets i_l$, $v\gets v_i$
\endchange

\bugonpage 4f5.222 in step R2' (20.02.05)
$p\ge L$ \becomes $w\ge L$
\endchange

\bugonpage 4f5.223 line 19 (20.04.09)
$(1+3wz^2+2w^2z^3+z^2z^4+w^3z^4)^8$ \becomes
$(1+3wz^2+2w^2z^3+w^2z^4+w^3z^4)^8$
\endchange

\bugonpage 4f5.224 replacement for lines 1--6 of answer 2 (20.10.24)
\ans2. (a) (Solution by X. Lou.) We can prove that
$\.{LLINK($a_k$)}=a_k-1$ and $\.{RLINK($a_k$)}=(a_k+1)\mod(n+1)$
when $a_k$ is undeleted; hence that undeletion sets
\.{RLINK($a_k-1$)} and \.{LLINK($(a_k+1)\mod(n+1)$)} to the correct
value~$a_k$. (If $a_k-1$ wasn't deleted {\it before\/} $a_k$,
\.{LLINK($a_k$)} never changed. Otherwise
\.{LLINK($a_k$)} became $a_k-1$ when $a_k-1$ was undeleted, by
induction on~$k$. A~similar argument works for \.{RLINK}.)
Notice that each \.{LLINK} and \.{RLINK} is reset exactly once, except
that \.{LLINK($1$)} and \.{RLINK($n$)} remain~0.\tighten\par
\endchange

\improvepage 4f5.226 line 2 of answer 12 (21.07.30)
\.{NAME(TOP($q$))} \becomes `$@$\.{NAME(TOP($q$))}$\?\?$'
\endchange

\amendpage 4f5.226 append new paragraph to answer 12 (20.03.27)
[Algorithm C extends Algorithm X to colors. If $\.{COLOR($q$)}\ne0$,
also print `\.{:$c$}' where
$c=\.{COLOR($q$)}$ if $\.{COLOR($q$)}>0$, otherwise $c=\.{COLOR(TOP($q$))}$.]
\endchange

\bugonpage 4f5.229 last line of answer 27 (21.08.06)
5 4 6  10 12 5 \becomes
5 4 6 8 10 12 5
\endchange

\bugonpage 4f5.229 last line of answer 29 (19.12.14)
$\{1,2\}$ \becomes $\{0,1\}$. [Also change
$\vcenter{\baselineskip6pt\sevenrm\halign{#\hfil\cr01\cr10\cr}}$ to
$\vcenter{\baselineskip6pt\sevenrm\halign{#\hfil\cr10\cr01\cr}}$ in
lines 4--5, 9--10, 13--14 of the matrix.]
\endchange

\amendpage 4f5.231 last line of answer 37 (21.03.03)
{\sl Combinatorics}, to appear. \becomes
{\sl Combinatorics\/ \bf27} (2020), \#P1.52, 1--27.
\endchange

\amendpage 4f5.231 replacement for step G3 in answer 38 (21.02.07)

\algindentset{G1}
\algstep G3. [Found?] If $k>n-r$ go to G4.
Otherwise if $a_k=b_{k+n}=c_{k-n}=0$, go to G5.
Otherwise set $k\gets k+1$ and repeat this step.
\endchange

\amendpage 4f5.234 replacement for answer 52 (19.12.14)
\ans52. (Solution by F. Stappers.)
Puzzles claiming to be ``the world's hardest sudoku'' keep appearing in
online forums. Rated by search tree size with Algorithm~X, the toughest
among nearly 27,000 such extreme puzzles
is shown here in a canonical form.
(It's number 6539 in a list available from
{\tt sites.google.com/site/sudoeleven/} (2011).)
% 1000 trials, seeds 314???
Its random\-ized search tree sizes
are $24400\pm1900$\dash---astonishingly
high for sudoku; and its mean running time is about 12~M$\mu$.)
{\hangindent-1in \hangafter0\tighten\parfillskip=0pt\hfil
\def\epsfsize#1#2{.85#1}%
\smash{\rlap{\quad\raise2pt\hbox{\epsfbox{\figdir/v4b.2898}}}}\par}
\endchange

\bugonpage 4f5.236 lines 6 and 7 (21.02.05)
There are only two solutions \dots\ symmetric. \becomes
Include `$\ast_j{:}k$' in option $(0,j,k)$.
There are only four solutions, found in 3~G$\mu$,
centrally symmetric and reducing under transposition to only two.
\endchange

\bugonpage 4f5.239 last line of answer 70 (21.05.25)
63]; \becomes 63;
\endchange

\bugonpage 4f5.239 line $-3$ of answer 72 (20.07.04)
1292687 packings \becomes 1292697 packings
\endchange

\amendpage 4f5.240 replacement for last line of answer 74 (21.12.18)
\indent [See {\tt www.solitairelaboratory.com/puzzlelaboratory/DominoGG.html}.] 
\endchange

\bugonpage 4f5.241 line 2 of answer 77 (20.03.02)
each pair of vertices \becomes each pair of edges
\endchange

\bugonpage 4f5.241 in the table for answer 80, to match page 87 (20.03.27)
\eightpoint
\.{LEN($x$)} \becomes \.{LEN($x$)},\thinspace\.{TOP($x$)}\nl
[the entries for \.{COLOR($x$)} in nodes $x=6, 11, 16, 19, 22, 25$ should
be zero, not `\hbox{---}'.]
\endchange

\bugonpage 4f5.241 line 2 of answer 83 (20.01.01)
$j>N$ \becomes $j>N_1$
\endchange

\amendpage 4f5.242 line 6 of answer 87 (21.05.25)
items~$w'$ and appending $c_1\ldots c_n{}'$ \becomes
items~$w\@$ and appending $c_1\ldots c_n\!\@$
\endchange

\bugonpage 4f5.243 in answer 90 (21.04.18)
line 12: \.{LOVED}\kern.5pt\.{\char`\|}\kern.5pt\.{GIVES} \becomes
         \.{LOVED}\kern.5pt\.{\char`\|}\kern.5pt\.{GIVEN}\nl
line 13: \.{BONES} \becomes \.{TONES}\qquad and\qquad
         \.{WHILE} \becomes \.{WHOLE}
\endchange

\amendpage 4f5.244 in answer 91 (21.06.06)
line $-8$: the shortest \becomes an optimum\nl
line $-5$: \.{FALLS} \becomes \.{WALLS}
\endchange

\amendpage 4f5.244 line 2 of answer 92 (21.05.25)
$w$, $w'$ \becomes $w$, $w\@$
\endchange

\amendpage 4f5.245 near the top (21.05.25)
line 1: $c_1\ldots c'_n$' \becomes $c_1\ldots c_n\!\@$'\nl
line 3: items $w'$ \becomes items $w\@$
\endchange

\advance\parindent by 4pt % for them BIG answer numbers
\bugonpage 4f5.248 line 1 of answer 104 (19.12.16)
We may \becomes (a) We may
\endchange

\improvepage 4f5.249 lines 6, 7, 10, 15, 16, 17 of answer 109 (21.05.25)
[enclose each option in single quotes; for example, line 6 should begin with
`$@\#k$ and line 7 should end with $\.0@$']
\endchange

\amendpage 4f5.250 new copy following the displayed solutions
  in answer 109 (21.09.16)
\indent Suppose there are $w$ words of total length~$s$. Aaron~Windsor suggests
adding options `E $ij$:\.. $ij'$:0' for $1\le i\le m$ and $1\le j\le n$,
where E~is a new primary item representing an empty cell. All solutions to
the {\mc MCC} problem with the number of E's in the interval
$[mn-s+w-1\dts mn]$ are then either
connected or contain a cycle.\par
\endchange

\bugonpage 4f5.253 replacement for line $-3$ (21.10.15)
\noindent
$$\def\epsfsize#1#2{.33#1}\vcenter{\epsfbox{\figdir/v4b.2920}}$$\par
\endchange

\bugonpage 4f5.256 line $-5$ (21.10.21)
\.{13:a} or \.{-13:c} \becomes
\.{-13:a} or \.{-13:c}
\endchange

\amendpage 4f5.259 replacement for $Q$ in answer 136, line 16 (20.10.27)
$\displaystyle Q=
{1\over2}\pmatrix{-1&-\phi&1/\phi\cr -\phi&1/\phi&-1\cr1/\phi&-1&-\phi\cr}$
\becomes
$\displaystyle Q=
{1\over2}\pmatrix{1&-\phi&1/\phi\cr \phi&1/\phi&-1\cr1/\phi&1&\phi\cr}$
\endchange

\bugonpage 4f5.260 line 9 of answer 138 (20.07.23)
$96/6=12$ cases \becomes $96/6=16$ cases
\endchange

\bugonpage 4f5.263 line $-4$ of answer 143 (21.10.21)
$(5,1,1,0)$ \becomes $(12,12,1,0)$
\endchange

\bugonpage 4f5.263  line 5 of answer 145 (20.07.24)
3 three odd \becomes 3 odd
\endchange

\bugonpage 4f5.277 replacement for answer 189{(a)} (19.12.19)
(a) $\vert e^{e^z}\vert=\vert e^{e^x\cos y+ie^x\sin y}\vert=
 \exp(e^x\cos y)$; $\vert e^{-e^z}\vert=\exp(e^{-x}\cos y)$.\par
\endchange

\amendpage 4f5.280 line $-4$ of answer 207 (20.11.06)
96 G$\mu$, with a 55-meganode \becomes
92 G$\mu$, with a 54-meganode
\endchange

\bugonpage 4f5.280 line 7 of answer 208 (21.11.17)
inserts $V_{i(j-1)}$, $H_{(i+1)j}$, $V_{i(j+1)}$, \becomes
inserts $V_{ij}$, $H_{(i+1)j}$, $V_{i(j+1)}$, $H_{ij}$
\endchange

\bugonpage 4f5.284 replacement for answer 232 (20.10.18)
\def\dol/{$\losup\$$}%
\ans232. No. Algorithm X\dol/ finds 96 solutions of minimum cost \$84;
but the true solution in Fig.~\fig74/(a) actually costs \$86
by this measure. The effects of 16 rounding errors,
each potentially changing the result by nearly~\$1, have
invalidated everything.
[Therefore the author used $\$\lfloor 2^{32}d(i,j)\rfloor$ when
preparing Fig.~\fig74/. This was safe, because the distance between the
first 8 solutions and the 9th was greater than 16\dash---in fact,
{\it much\/} greater, although a difference of only 17 would have been
convincing.]\tighten
\endchange

\bugonpage 4f5.286 in answer 239 (19.11.26)
line 3: cost $\epsilon^i$ \becomes cost $\epsilon^j$\nl
line 9: {\sl Graphs and Algorithm\/} \becomes {\sl Graphs and Algorithms\/}
\endchange

\improvepage 4f5.286 line 8 of answer 239 (19.11.28)
smaller cost $\epsilon^4$ \becomes smaller additional cost $0+\epsilon^4$
\endchange

\bugonpage 4f5.288 line 3 of answer 249 (20.01.26)
Set $l=t=0$. \becomes Set $l\gets t\gets0$.
\endchange

\bugonpage 4f5.288 line 4 of answer 249 (21.12.08)
$p\gets t$, $r\gets1$ \becomes
$p\gets t$, $y\gets0$, $r\gets1$
\endchange

\bugonpage 4f5.293 lines 1 and 2 (20.08.28)
exact problem \becomes exact cover problem
\endchange

\bugonpage 4f5.294 line 5 of answer 270 (21.12.19)
90 rotated \becomes $90^\circ$-rotated
\endchange

\bugonpage 4f5.294 line 2 of answer 271  (21.12.19)
1152 \becomes 152
\endchange

\bugonpage 4f5.298 line 2 of answer 283 (20.09.02)
365(b) \becomes 282(b)
\endchange

\amendpage 4f5.302 line $-5$ of answer 298 (21.05.25)
{\sl Supplement\/} (1936) \becomes
{\sl Supplement\/ \bf2},\thinspace16 (February 1936)
\endchange

\bugonpage 4f5.304 replacement for first paragraph of answer 303 (20.02.06)
\ans303. (a) Represent the tree as a sequence $a_0a_1\ldots a_{2n-1}$ of
nested parentheses; then $a_0$ will match $a_{2n-1}$. The left
boundary of the corresponding parallomino is obtained by mapping each `\.('
into N or~E, according as it is immediately followed by `\.(' or `\.)'.
The right boundary, similarly, maps each `\.)' into N or~E according as it
is immediately {\it preceded\/} by `\.)' or `\.('. For example, if we take
7.2.1.6--\eq(1) and enclose it in an additional pair of parentheses,
the corresponding parallomino is shown below with part~(d).\tighten\par
\endchange

\bugonpage 4f5.304 replacement for lines 6, 7, 8 of part {(b)} (20.01.19)
\noindent $1+1+1+1+2+2+2+2+2+2+2+2+2+1+1$; there's an ``outer''~$G$,
whose $H$ is $yxyGy$, and an ``inner''~$G$, whose $H$ is $xyyxyxxy$.)
Thus we can write $G$ as a continued fraction,\tighten
$$G(w,x,y)=wxy\big/\bigl(1-wx-wy-w^3\?xy/(1-w^2\?x-w^2\?y-w^5\?xy/
   (\,\cdots\,))\bigr).$$
\endchange

\bugonpage 4f5.306 four lines after the first display (21.08.21)
[begin a new paragraph with the answer to part (e)]
\endchange

\bugonpage 4f5.313 lines $-3$ and $-2$ of answer 322 (21.05.25)
Dawson in the 1940s, \becomes Dawson,
\endchange

\amendpage 4f5.314 line 19 of answer 325 (21.11.11)
Section 7.1.4 \becomes Section 7.1.4.2
\endchange

\amendpage 4f5.314 last lines of answer 325 (21.06.30)
line $-2$: five special \becomes seven special\nl
line $-1$: B5f, R7d \becomes B5f, W4e, W2f, R7d
\endchange

\amendpage 4f5.315 in last paragraph of answer 327 (19.12.22)
Hall (clip, underpass) \becomes Hall (piano, clip, underpass)\nl
Morgan (face, gorilla, smile) \becomes Morgan (piano, face, gorilla, smile)
\endchange

\amendpage 4f5.318 line $-6$ of answer 337 (19.12.14)
[these illustrations should be darker, I've now thickened the lines]
\endchange

\bugonpage 4f5.322 line 7 (19.12.25)
{\sl Budapestiensis\/} \becomes {\sl Budapestinensis\/}
\endchange

\amendpage 4f5.322 lines $-3$ and $-2$ of answer 346 (19.12.14)
(online 18 June 2018) \becomes {\bf61} (2019), 271--284
\endchange

\amendpage 4f5.322 replacement for line $-2$ of answer 349 (20.10.18)
\noindent
$$\def\epsfsize#1#2{.27#1}
\def\\#1{\vcenter{\epsfbox{\figdir/v4b.72#1}}}
\let\quad=\enspace
\centerline{$\\0\quad\\0\quad\\0\quad\\0\quad\\1\quad\\2\quad\\3\quad
\\4\quad\\5\quad\\6\quad\\7\quad\\8\quad\\8\quad\\8\quad\\8$}$$\par
\endchange

\amendpage 4f5.322 replacement for line $-3$ of answer 350 (20.10.18)
\noindent
$$\def\epsfsize#1#2{.4#1}
\def\\#1{\vcenter{\epsfbox{\figdir/v4b.71#1}}}
\let\quad=\enspace
\\0\quad\\0\quad\\0\quad\\1\quad\\2\quad\\3\quad
\\4\quad\\5\quad\\6\quad\\7\quad\\7\quad\\7$$
\endchange

\bugonpage 4f5.325 line 13 (20.09.25)
planar places \becomes planar pieces
\endchange

\bugonpage 4f5.326 line 4 of answer 357 (20.09.28)
if and only the \becomes if and only if the
\endchange

\amendpage 4f5.330 and 331, replacement for answer 372 (21.09.18)
\ans372. (a) This fact, and the others noted below, can be proved by
induction on the number of rooms: If the lower right corner of the upper left
room is a $\bot$ junction, we can ``flatten'' and remove that room by bringing
its right bound left; otherwise we can bring its bottom bound up.
All floorplans can be built up by reversing this flattening process.\tighten\par
Let the rooms be $r_1\ldots r_n$ in diagonal order and $r_{p_1}\ldots r_{p_n}$
in antidiagonal order (left to right). Then $r_i\Downarrow r_j$
$\iff$ $i<j$ and $i$ follows~$j$ in the permutation $p=p_1\ldots p_n$;
$r_i\Rightarrow r_j$ $\iff$ $i<j$ and $i$ precedes~$j$ in~$p$. The
number of horizontal bounds is the number of {\it descents\/} in~$p$,
plus~2. The number of vertical bounds is the number of {\it ascents},
plus~2.\tighten\par
(b) Here's the twin tree structure for the example. Notice that its
leftward and rightward chains are the ordered sequences
of rooms adjacent to the bounds.
$$\def\epsfsize#1#2{.6#1}
\def\\#1{\vcenter{\epsfbox{\figdir/v4b.372#1}}}
\\4\qquad\\5$$\par
Every twin tree structure arises in a remarkably simple way:
Let $p=p_1p_2\ldots p_n$ be any permutation of $\{1,2,\ldots,n\}$.
Obtain $T_0$ by inserting $p_1$, $p_2$, \dots,~$p_n$ into an initially empty
binary tree; obtain $T_1$ similarly by inserting $p_n$, \dots, $p_2$, $p_1$.
Those trees can be constructed in linear time (exercise 6.2.2--50);
and it's easy to see that they are twins, both with inorder $12\ldots n$.
Although different permutations can yield the same twin tree,
exactly one {\it Baxter permutation\/} (exercise \MPR--135)
does so; and it can be computed from the twin tree in linear time(!).
Thus there are nice one-to-one correspondences between floorplans, twin trees,
and Baxter permutations.\par
[Floorplans are important in VLSI layout, where rooms
correspond to modules and bounds correspond to channels.
Twin trees were introduced by S.~Dulucq and O.~Guibert in
{\sl Discrete Math.\ \bf157} (1996), 91--106, purely for their combinatorial
interest, then applied to floorplans by B.~Yao, H.~Chen, C.-K.~Cheng,
and R.~Graham in {\sl ACM Trans.\ Design Aut.\ Electronic Syst.\ \bf8}
(2003), 55--80. See also J.~M. Hart, {\sl Int. J. Comp. Inf. Sciences\/ \bf9}
(1980), 307--321;
 H.~Murata, K.~Fujiyoshi, T.~Watanabe,
and Y.~Kajitani, {\sl Proc.\ Asia South Pacific Design Aut.\ Conf.\
\bf2} (1997), 625--633;
E.~Ackerman, G.~Barequet, and R.~Y. Pinter, {\sl Discrete
Applied Mathematics\/ \bf154} (2006), 1674--1684;
and the author's programs
{\mc FLOORPLAN-TO-TWINTREE},
{\mc TWINTREE-TO-BAXTER},
{\mc BAXTER-TO-FLOORPLAN}, available online (2021).]\tighten
\endchange

\amendpage 4f5.332 and 333, in answer 375 (20.02.10)
perimeter \becomes semiperimeter \quad [throughout]
\endchange

\bugonpage 4f5.333 in answer 375{(b)} (20.02.10)
$h_6,h_4)$ \becomes $h_6,h_5)$\qquad [four times]
\endchange

\bugonpage 4f5.333 line 4 of answer 376 (20.02.09)
$d=(w+1)/(w(w+2))$ \becomes $d=(w+2)/((w+1)(w+3))$
\endchange

\improvepage 4f5.333 line 3 of answer 378 (21.08.21)
has a subset \becomes has a finite subset
\endchange

\bugonpage 4f5.334 in answer 379 (21.05.25)
line 2: $h\times(w-2)$ \becomes $5\times(w-2)$\nl
lines 6 and 7: or ($h=5$ and $w$ is odd) or \becomes or
\endchange

\improvepage 4f5.335 line 4 of answer 383 (19.11.30)
dissection, into \becomes
dissection, of a $92\times92\times92$ cube into
\endchange

\amendpage 4f5.336 lines 5 and 8 (21.05.25)
[these illustrations should be darker, I've now thickened the lines]
\endchange

\amendpage 4f5.340 last line of answer 398 (20.03.30)
published by Tatsuo Yano \becomes published by Ryuoh Yano
\endchange

\amendpage 4f5.341 lines 2 and 5 of answer 401 (21.05.25)
[these illustrations should be darker, I've now thickened the lines]
\endchange

\amendpage 4f5.344 replacement for answer 413 (19.12.23)
\ans413. (Solution simplified by R. Molinari.)
Let each record for an item include two new fields \.U and \.V.
The \.U and \.V fields of a secondary item that represents
edge $u\adj v$ will point to the primary items $u$ and~$v$. The \.U and \.V
fields of a primary item that represents vertex~$v$ are renamed \.{MATE}
and \.{INNER}. \.{MATE($v$)} is zero until $v$ first becomes the endpoint
of an edge, after which it points to the other endpoint of the path
fragment containing that edge. \.{INNER($v$)} is nonzero when $v$~lies
within a path fragment.\par
Introduce two new global variables:
Global variable \.F is the current number of fragments. Global variable
\.E is the edge that closed a loop, or zero if there's no loop.\par
For example, suppose two edges currently have color~1, say
$v_1\adj v_2$ and $v_3\adj v_4$. Then we've set
$\.{MATE($v_1$)}\gets v_2$,
$\.{MATE($v_2$)}\gets v_1$,
$\.{MATE($v_3$)}\gets v_4$,
$\.{MATE($v_4$)}\gets v_3$, and $\.F\gets2$.
If now $v_2\adj v_5$ joins the fray, we set
$\.{MATE(MATE($u$))}\gets u$,
$\.{MATE(MATE($v$))}\gets v$, and
$\.{INNER($v_2$)}\gets1$;
but we leave \.{MATE($v_2$)} unchanged.
Subsequent edges to~$v_2$ are rejected.\tighten\par
When the `purify' routine \eq(55) is called to give color~1 to a new edge~$i$,
it will refuse to do~so when \.E is nonzero, because a loop has already
been closed. Furthermore, when $\.E=0$,
it will know that edge~$i$ shouldn't be chosen if
\.{U($i$)} and \.{V($i$)} are mates and $\.F\ne1$,
because that would close a loop disjoint from other fragments.
On the other hand, it {\it will\/} close the loop if $\.F=1$,
also setting $\.E\gets i$.\par
All of these operations are nicely and easily undone
when we need to `unpurify'.
For example, suppose edge~$i$ loses color~1 when $u=\.{U($i$)}$
and $v=\.{V($i$)}$. If $v=\.{MATE($u$)}$, we unclose the loop
(and set $\.E\gets0$) if $i=\.E$;
otherwise we zero the mates and set $\.F\gets\.F-1$.
If $\.{MATE($u$)}\ne\.{MATE($v$)}$, we set
$\.{MATE($u$)}\gets u$,
$\.{MATE($v$)}\gets v$,
$\.{INNER($u$)}\gets\.{INNER($v$)}\gets0$, $\.F\gets\.F+1$.
The case $\.{MATE($u$)}=\.{MATE($v$)}$ is easy too.\par
\em Caution: Algorithm P must be modified so that it never discards redundant
items, when it is used to preprocess a problem for this extension
of Algorithm~C.
\endchange

\amendpage 4f5.346 additional material for answer 416 (21.12.21)
But $\smash{\underline{\beta(2)}}$ remains unknown. \becomes\par\smallskip
The intriguing case of $\smash{\underline{\beta(2)}}$ remains unknown.
Beluhov proved that it is at most $11\over16$, using a construction for
$n=4k$ that's illustrated here for $n=\nobreak12$.
Palmer Mebane has contructed a 2-homogeneous puzzle on an $8\times8$ board
that has only 24 clues [see\break
{\tt puzsq.jp/main/puzzle\char`\_play.php?pid=14178}].%
{\hangindent-11pc\hangafter0 \parfillskip=0pt\hfil\tighten
 \smash{\rlap{\qquad
  \lower1pt\vbox{\def\epsfsize#1#2{.45#1}\epsfbox{\figdir/v4b.1987}}\qquad
  \raise10pt\vbox{\def\epsfsize#1#2{.45#1}\epsfbox{\figdir/v4b.1985}}}}\par}
\endchange

\amendpage 4f5.346 replacement for answer 420 (21.12.20)
\ans420. (Solution by Palmer Mebane.) In any solution, each cell is either
inside or outside the loop. {\sl Lemma: Every\/} \.2 {\sl has exactly two neighbors
inside.} (For if the \.2 is outside, the neighbors opposite its two edges
are inside; otherwise the neighbors opposite its two {\it non\/}edges
are inside.) Let $S$ be the set of cells next to a~\.2. Color each~\.2
alternately red or blue; then each cell of~$S$ is a neighbor of one red~\.2
and one blue~\.2. In particular, that's true for each {\it inside\/} cell
of~$S$. Thus, by the lemma, there are equally many red and blue cells.
But that contradicts $m\mod4=n\mod4=1$!\par
[This exercise is due to N.~Beluhov, who observes that solutions aplenty
exist when $m$ is odd and $n\mod4=3$.]
\endchange

\amendpage 4f5.346 {(originally page 347)} line $-3$ of answer 421 (20.03.30)
by Tatsuo Yano \becomes by Ryuoh Yano
\endchange

\bugonpage 4f5.346 {(originally page 347)} last line of answer 421 (19.12.24)
{\bf89} \becomes {\bf90}
\endchange

\amendpage 4f5.348 replacement for lines 3--5 of answer 425 (19.12.25)
\noindent not.
Solutions for $k=2$ exist for all $n\ge3$; solutions for $k=3$ exist for
all $n\ge4$; solutions for $k=4$, due to B.~S. Ho, exist for all $n\ge5$;
solutions for $k=5$ and $k=6$, due to G.~J.~H. Goh, exist for all $n\ge6$.
(See (xviii)--(xxiv) in Fig.~\fig A--5/.) Goh has also discovered
analogous constructions for $k=7$, 8, 9,~10.
\endchange

\improvepage 4f5.348 lines 4 and 5 of answer 426 (20.03.30)
\setbox1=
\hbox{$\def\epsfsize#1#2{.75#1}\,\vcenter{\epsfbox{\figdir/v4b.951}}\,$}%
\setbox0=
\hbox{$\def\epsfsize#1#2{.75#1}\,\vcenter{\epsfbox{\figdir/v4b.950}}\,$}%
there cannot be three consecutive `\copy1',  nor five
`\copy0's that form an X or~T. \becomes
there cannot be three consecutive `\copy1's.
\endchange

\bugonpage 4f5.349 replacement for {(xx), (xxi), (xxii)} in Fig.~A--5 (19.12.24)
\begingroup
\def\epsfsize#1#2{.7#1}%
\noindent\epsfbox{\figdir/v4b.4920}\qquad
         \epsfbox{\figdir/v4b.4921}\qquad
         \epsfbox{\figdir/v4b.4922}
\endgroup
\endchange

\amendpage 4f5.353 last line of answer 438 (21.11.11)
Section 7.1.4 \becomes Section 7.1.4.2
\endchange

\bugonpage 4f5.355 line 14 of answer 445 (19.12.04)
an {\mc MCC problem} \becomes an {\mc MCC} problem
\endchange

\amendpage 4f5.356 three new lines for the end of answer 449 (20.02.07)
\noindent
The current record for largest literary hitori nugget, $12\times5=60$,
was found by
Gary McDonald in September 2019: ``Ruth intimated that, as far as
she could judge, he was a very eligible swain.'' [Charles Dickens,
{\sl Martin Chuzzlewit}.]
\endchange

\amendpage 4f5.359 two new answers for exercise 416 (21.12.21)
\noindent
$$\def\epsfsize#1#2{.45#1}
\vcenter{\epsfbox{\figdir/v4b.1988}}\qquad
\vcenter{\epsfbox{\figdir/v4b.1986}}$$
\endchange

\amendpage 4f5.360 include a missing answer (21.05.25)
\centerline{\def\epsfsize#1#2{.54#1}$\vcenter{\epsfbox{\figdir/v4b.3337}}$\quad
(see answer 431)}
\endchange

\expandafter\ifx\csname indexeject\endcsname\relax\else\vfill\eject\fi
\amendpage 4f5.361 and following (19.09.17)
Miscellaneous changes to the existing index of Volume~4 Fascicle~5
are collected here,
including corrections and amendments to the old entries as well as new entries
that are occasioned by the new material. Thus, the lines of the full index
that have changed serve also as an index to the present document. However,
when a correction or amendment has caused an old index entry to be deleted,
the deletion is usually not indicated.

\input exotic
\begindoublecolumns
\begingroup
\indexformat
\gdef\Uni1.08:{\bitmap24:1.08:}
\hangindent2em
$\implies$: Implies. % 3rd
$\iff$: If and only if. % 3rd
Ackerman, Eyal ({\heb\Hfn/\Hmm/\Hrr/\Hqq/\Haa/ \Hll/\Hyy/\Hyy/\Haa/}), 331. % 3rd
Agarwal, Akshay Kumar ({\dn a\kern0.1em{\char34}y k\kern-0.4em\char0\kern.45emmAr ag\llap{\char125}vAl}), 207. % 3rd
Ascents of a permutation, 330, 384. % 3rd
Ase, Mitsuhiro (\JC46:48:-2:40% J1604
<c00000000000f01c00000000fffe00000030fffe00000078f03efffffffcf03e7ffffffc%
 f03e00000f00f03e00000f00f03c00000f00f03c00000f00f07800000f00f07000000f00%
 f0e030060f00f0c03c0f0f00f1803fff8f00f3003fff8f00f6003c0f0f00f6003c0f0f00%
 f3803c0f0f00f1c03c0f0f00f0e03c0f0f00f0e03c0f0f00f0703c0f0f00f0783c0f0f00%
 f0783c0f0f00f03c3c0f0f00f03c3c0f0f00f03c3c0f0f00f03c3c0f0f00f03c3fff0f00%
 f03c3fff0f00f03c3c0f0f00f03c18060f00f07c00000f00f7fc00000f00f1f800000f00%
 f0f800000f00f07000000f00f00000000f00f00000000f00f00000000f00f00000000f00%
 f00000000f00f00000001f00f0000007ff00f0000000fe00f00000007e00600000003c00%
 >%% unicode char "963f
\JC48:48:0:40% J3205
<1c003000000c0f003c00001e07c03e03ffff03e03e01ffff01f03c0001e001f03c0003e0%
 00f03c3003c000603c780780000ffffcc7180007fffcee3c00003c00fffe00003c00fffe%
 c00c3c20f03c700e3c70f03c780ffff8f03c3c0ffff8f03c3e0e3c70f03c3e0e3c70f03c%
 3e4e3c70fffc1c4e3c70fffc004e3c70f03c004e3c70f03c004e3c70f03c00ce3c70f03c%
 00ce3c70f03c00ce3c70f03c00cffff0f03c01cffff0fffc01ce3c70fffc018c3c20f03c%
 03803c00f03c03803c00f03c07807f00f03c1f007f80f03cff007fc0f03c7f00fde0f03c%
 3f00fcf0fffc3e01fcf0fffc1e03fcf060181e03bc6000001f073c0010e01f0e3c003878%
 0f9c3c003c3e0fb03c007c1f0fc03c00780f0fc03c01f00f07c03c07e00f0380181f0006%
 >%% unicode char "702c
\JC48:48:0:40% J2487
<000007800000000007c00000000003e00380000003e003e0038003c003f003c003c003f0%
 01f003c003e001fc03c003e000fe03c007c0007e03c00780003f03c00f00003f03c00e00%
 001f03c01c00000f03c03800000f03c07000000603c0e000000003c3c000000003c30000%
 000003c0000c000003c0001effffffffffff7fffffffffff0000f03c00000000f03c0000%
 0000f03c00000000f03c00000000f03c00000000f03c00000000f03c00000000f03c0000%
 0000f03c00000000f03c00000001f03c00000001f03c00000001f03c00000001e03c0000%
 0003e03c00080003e03c000c0007c03c000c0007c03c000e000f803c000f001f003c000f%
 003e003c000f007c003e001f00f8003fffff07e0003fffffffc0001ffffefe00000ffffc%
 >%% unicode char "5149
\JC46:48:-2:40% J3572
<00000c00000000000f00000000000f80000000000f80000004000f00000004000f000000%
 0c000f0000300c000f0000781ffffffffffc1ffffffffffc3c00000000787c0000000078%
 fc00000000f0f8000c0001e0f8000f0003c070000f80038000000f80000000000f000000%
 00000f00000000000f00000003000f00180003c00f003c0003fffffffe0003fffffffe00%
 03c00f003c0003c00f003c0003c00f003c0003c00f003c0003c00f003c0003c00f003c00%
 03c00f003c0003c00f003c0003fffffffc0003fffffffc0003c00f003c0003c00f003c00%
 03c00f003c0003c00f003c0003c00f003c0003c00f003c0003c00f003c0003c00f003c00%
 03fffffffc0003fffffffc0003c000003c0003c000003c0003c000001800018000000000%
 >%% unicode char "5b99
), 346. % 2nd
Asymptotic methods, 11, 12, 16, 26, 53, 145--146, 226, 231, 232. % 3rd
Aztec diamonds, 153, 155, 290. % 3rd
Barequet, Gill ({\heb\Htt/\Hqq/\Hrr/\Hbb/ \Hll/\Hyy/\Hgg/}), 331. % 3rd
Baumert, Leonard Daniel, 52, 221, 371. % 3rd
Baxter, Glen Earl, 384. % 3rd
\sub permutations, 27, 330. % 3rd
Binary trees, 170. % 3rd
Binomial coefficients, 384. % 3rd
Bitwise {\mc AND} ($\band$), 17, 126, 208, 227, 350. % 2nd
Bitwise {\mc OR} ($\bor$), 17, 126. % 2nd
Bounds and rooms, 169--170, 384. % 3rd
Boyce, William Martin, 384. % 3rd
Boyd, Stephen Poythress, 189, 207. % 3rd
Briggs, Preston, 39. % 3rd
Brotchie, Alastair, 245. % 2nd
Bucket elimination, \see Frontier. % 2nd
Chen, Hongyu (\GC73:79:-6:64% G1934
<000000000060000000000000000000780000000000000000007c0000000000000000007f%
 0000000000000000007f80000000e0003800007f80000000f800fc00007f00000000fe01%
 fe0000fe00000000ffffff0000fc00000000ffffff8001f800000000ffffff8001f00000%
 7000fc01ff8003f00000f800fc01ff8007e00001fc00fc01fe000fc00003fe00fc03ffff%
 ffffffffff00fc03f3ffffffffffff80fc03f1ffffffffffff80fc07e0801f8000000000%
 fc07e0003f0000000000fc07c0003f0000000000fc07c0007e0000000000fc0f80007e00%
 00000000fc0f8000fe0e00000000fc0f8000fc0f80000000fc1f0001fc07e0000000fc1f%
 0001fc07f8000000fc1e0003f807f8000000fc3e0003f807f0000000fc3c0007f007e000%
 0000fc3e0007f007e0000000fc1f000ff007e0000000fc0f800fe007e0000000fc07c01f%
 e007e0000000fc07e01fc007e0000000fc03f01f8007e001c000fc01f81f0007e003e000%
 fc00f81f0007e007f000fc00fc7e0007e00ff000fc007ffffffffffff800fc007fffffff%
 fffffc00fc003ffffffffffffc00fc003fbe0007e0000000fc001fdc0007e0000000fc00%
 1fc00007e0000000fc001fc06007e0000000fc001fc0f007e0000000fc000fc0fc07e000%
 0000fc000fc0fe07e0000000fc000fc1ff07e0000000fc000fc1ff07e0000000fc001fc1%
 fe07e0000000fc001f83fc07e1c00000ffffbf83f807e1f00000ffffff07f007e0f80000%
 fc7fff07e007e07e0000fc3ffe0fc007e03f0000fc0ffe1f8007e01fe000fc07fc1f8007%
 e00ff800fc07f83f0007e007fe00fc07f03e0007e003ff00fc04007c0007e001ff00fc00%
 00fc0007e000ff80fc0001f80007e000ff80fc0003f00007e0007f80fc0003e00007e000%
 3f80fc0007c00007e0001f80fc000f800007e0001f80fc001f000007e0000f80fc003e00%
 0007e0000f80fc007c000007e0000780fc00f800ffffe0000000fc01e000ffffe0000000%
 fc03c0001fffe0000000fc07800001ffc0000000fc06000000ffc0000000fc040000007f%
 80000000fc000000007e00000000f8000000007c00000000f8000000002000000000>% uni9648
\JC48:48:0:40% J2508
<000007000000000007800000000003e00000000003e00000030003c00000030003c00000%
 038003c0000c078003c0001e07ffffffffff0fffffffffff0f000000001f1f000000003e%
 1f000e00003c1f000f00003c1e000fc000780c000fc000e000000f80000000000f000000%
 00000f00000c00000f00001effffffffffff7fffffffffff00001e00000000001e000000%
 00003e30000000003c3c000000003c3f00000000781f00000000781e00000000f01e0000%
 0001f01c00000001e03c00000003c03800000007c0780000000f80703000000f00f03c00%
 001e00e01e00003c01c01f00007803c00f8000f0038007c001c007001fe007800e03ffe0%
 3e0f1e7ffff03c0ffffff1f00007fffc01f00007ffc001f00007f00001f00003000000e0%
 >%% unicode char "5b8f
\JC48:48:0:40% J1707
<000003000000000003c00000010003e00000010003e00000038003c0000c038003c0001e%
 07ffffffffff0fffffffffff1f800000003e1f800000003c3f00000000383f0000000070%
 3e00000000601800000000c0000000000c00000000001e0000ffffffff00007fffffff00%
 000003c00000000003c00000000003c00000000003c00000000003c00000000003c00000%
 000003c0000c000003c0001effffffffffff7fffffffffff000003c00000000003c00000%
 000003c00000000003c00000000003c00000000003c00000000003c00000000003c00000%
 000003c00000000003c00000000003c00000000003c00000000003c00000000003c00000%
 000003c00000000007c0000000007fc0000000001fc0000000000f800000000007000000%
 >%% unicode char "5b87
), 331. % 3rd
Cheng, Chung-Kuan (\JC46:48:-2:40% J3636
<c00c00700000e01e007c0000ffff003e0000ffff003c0000f01e003c0030f01e003c0078%
 f01ffffffffcf03efffffffcf038003c0000f038003c0000f070803c0200f070c03c0700%
 f0e0ffffff80f1c0ffffff80f300f03c0f00f380f03c0f00f0c0f03c0f00f060f03c0f00%
 f070f03c0f00f030f03c0f00f038ffffff00f038ffffff00f03cf03c0f00f03cf03c0f00%
 f03cf03c0f00f03cf03c0f00f03cf03c0f00f03cf03c0f00f03cffffff00f03cffffff00%
 f03c607c0600f03c00fc0000f3fc00fe0000f1fc00ff0000f0f801ff8000f07001fdc000%
 f00003bde000f00007bcf000f0000f3c7800f0001e3c3e00f0003c3c1f80f000783c1ffc%
 f001f03c0ffcf003e03c07f8f03f803c03f8f03c003c00f0f000003c0000600000180000%
 >%% unicode char "9673
\GC65:79:-8:63% G5448
<0000001e00000000000000001f80000000000000000ff0000000000000000ff000000000%
 0000000fe0000000000000000fc0000000000000000f80000000000000000f8000000000%
 0000000f80000000000000000f80000000000000000f80000000000000000f8000000000%
 0000000f80000000000000000f80000000000000000f80000000000000000f8000007800%
 f000000f800000fc00fe00000f800001fc007ffffffffffffffe007fffffffffffffff00%
 7fffffffffffffff807e00000f800000fc007e00000f800000fc007e00000f800000fc00%
 7e00000f800000fc007e00000f800000fc007e00000f800000fc007e00000f800000fc00%
 7e00000f800000fc007e00000f800000fc007e00000f800000fc007e00000f800000fc00%
 7e00000f800000fc007e00000f800000fc007e00000f800000fc007e00000f800000fc00%
 7e00000f800000fc007e00000f800000fc007e00000f800000fc007e00000f800000fc00%
 7e00000f800000fc007ffffffffffffffc007ffffffffffffffc007ffffffffffffffc00%
 7e00000f800000fc007e00000f800000fc007e00000f800000fc007e00000f800000fc00%
 fe00000f800000fc00fe00000f800000fc00fe00000f800000fc00fc00000f8000000000%
 0000000f80000000000000000f80000000000000000f80000000000000000f8000000000%
 0000000f80000000000000000f80000000000000000f80000000000000000f8000000000%
 0000000f80000000000000000f80000000000000000f80000000000000000f8000000000%
 0000000f80000000000000000f80000000000000000f80000000000000000f8000000000%
 0000000f80000000000000000f80000000000000000f80000000000000000f8000000000%
 0000000fc0000000000000001fc0000000000000001fc0000000000000001fc000000000%
 0000001fc0000000000000001f80000000000000001f8000000000>%% unicode char "4e2d
\KC
<000000000000000000000000000000000000000000000000000000000000080000000000%
 0000000f000000000000000003c00000000000000001f00000000000400000f000000000%
 00c000007800000c0000c000003000001f0000c000000000003f8000ffffffffffffffc0%
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 0007f00000ffffffe000ff8000001ffffe000ffe000000000000001fc000000000000000%
 000000000000000000000000000000000000000000000000000000000000000000000000%
>%% Unicode char "5bec
), 331. % 3rd
Chung Graham, Fan Rong King (\JC48:48:0:40% J3065
<01c00000018001f00000078001fc00003fc003fe0001ffe003f8000fff0003fe01fff000%
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 3dee03fffffc3de8e1fffffc39e38000f00001ff0000f00003fc0000f0003ff80000f000%
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 >%% Unicode char "937e
\GC74:71:-3:61% G2980
<00000000e0000000000000000001fc000000000000000001ff000000000000000003ff80%
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\GC73:74:-4:61% G2328
<000003c0000780000000000003f00007e0000000000001fc0003f8000000000001fe0003%
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 8000000000070000000000000000000400000000>%% Unicode char "82b3
\GC73:75:-5:62% G4056
<00000f0000000000000000000fc0000f00000000000007f8000fe0000000000007fc000f%
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), 384. % 3rd
Chuzzlewit, Martin, 356. % 2nd
Coffin, Stewart Temple, 324. % 3rd
Connected components, 134, 142--143, 151, 162, 167, 244, 248, 250. % 3rd
Convex functions, 4, 8, 16, 20, 27, 189, 194, 195. % 3rd
Cutsets, 344. % 2nd
$d^+(v)$ (out-degree of $v$), 218, 247, 337. % 2nd
Data structures, 30--32, 35, 37--40, 44, 56, 63--67, 94--95, 107, 118, 170. % 3rd
Davis, Horace Chandler, 184. % 3rd
Descents of a permutation, 330, 384. % 3rd
Determinants, 384. % 3rd
Dickens, Charles John Huffam, 356. % 2nd
Distance, dynamically updating, 57. % 2nd
Downloadable programs, ii, v, 331. % 3rd
Dulucq, Serge, 331, 384. % 3rd
Embedded graphs, 130. % 3rd
Entropy, 24, 25, 27. % 3rd
$f^\uparrow$ (maximal elements of family~$f$), 353--354. % 2nd
Floorplans, 169--170, 384. % 3rd
Franel, J\'er\^ome, 207. % 2nd
Fujiyoshi, Kunihiro (\JC48:48:0:40% J3803
<0000c00c00000000f00f00000000f80f800c0000f00f001effffffffffff7fffffffffff%
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 380f0f878fe0381f060787e031ff043f83e0707e001f81c0603e000f8000c01c00070000%
 >%% unicode char "85e4
\JC48:48:0:40% J2140
<000003000000000003c00000000003e00000000003e00000000003c00000000003c00000%
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 >%% unicode char "5409
\JC45:48:-2:40% J4314
<003800180060003e001e00f0003e001ffff8003c001ffff8003c001e00f0003c001e00f0%
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 >%% unicode char "90a6
\JC48:48:0:40% J4546
<03800000020001e03e00030001f00f8007c000f807e007f0007c03f007e0007c01f80fc0%
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 0fc00003c0000fc00003c0000fc00003c0000fc00003c00007c00003c000078000018000%
 >%% unicode char "6d0b
), 331. % 3rd
Games, 8, 13. % 3rd
Goh, Jun Herng Gabriel (\GC76:74:-3:60% G4666
<00000000000001c000000000f000000007e0000000007c0000000fe0000000007fffffff%
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\GC79:77:-1:63% G3094
<000000000001e0000000000000000001f8000000000000000003fe000000000000000003%
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 000002000001ffc0000000000000>%%  unicode char "5cfb
\GC74:74:-3:61% G2667
<001c0000000000000000001f8000000000000000001fe000000000000000001ff0000000%
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 001f8000000000000000001e0000000000000000>%% Unicode char "6052
), 348. % 2nd
Golomb, Solomon Wolf, 35, 52, 77, 78, 82, 155, 221, 294, 310, 371. % 3rd
Graham, Ronald Lewis (\GC72:74:-4:61% G2480
<0000007c000f8000000000007f800ff000000000003fc00ff000000000003f000fc000e0%
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 000000000000038000000000000000020000>%% Unicode char "845b
\\\GC75:65:-3:60% G3302
<00000003e0000000000000000001f8000000000000000001fe000000000000000000ff00%
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\JC48:48:0:40% J5581
<00fc0000000000f80000000000f00000006000f0000000f000f0fffffff800f07ffffffc%
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 >%% Unicode char "6046
), 182, 328, 331, 364, 384. % 3rd
Guibert, Olivier, 331, 384. % 3rd
Hart, Johnson Murdoch, 331. % 3rd
Historical notes, 28, 31--32, 51--52, 79--80, 121, 198--200, 210, 226, 248, 258--262, 265, 277, 290--291, 293, 295, 301, 308, 317--319, 322, 325--326, 334, 337, 340, 343, 346. % 2nd
Ho, Boon Suan (\GC74:73:-4:60% G2646
<0000f0000000000000000000f8000000000000000000fe000000000000000001ff000000%
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\GC73:72:-4:61% G4636
<0000000380000000000000000001e0000000000000000000f0000000000000000000fc00%
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 0000000001f0000fe0000000000001f8000fe0000000000000f8001fe0000000000000fc%
 001fc0000000000000fc001fc0000000000000fc001fc00000000000007e003f80000000%
 0000007e003f800000000000003f007f000000000000003f007f000000000000003f007e%
 000000000000001f80fe000000000000001f80fc000000000000001fc1fc000000000000%
 000fc3f8000000000000000fe7f8000000000000000fe7f00000000000000007ffe00000%
 000000000007ffc00000000000000003ffc00000000000000003ff800000000000000001%
 ff800000000000000001ffc00000000000000001fff00000000000000003fff800000000%
 00000007fffc000000000000000ff7fe000000000000001fe3ff800000000000003fc1ff%
 c0000000000000ff807ff0000000000001ff003ff8000000000007fe001ffe0000000000%
 0ff8000fffe0000000003fe00007fffe00000000ffc00001ffffff000007ff0000007fff%
 ff80001ffc0000001fffff80007ff000000007fffc0003ffe000000001fff8001fff8000%
 0000000ff000fffe0000000000004000fff00000000000000000ffc00000000000000000%
 >%% Unicode char "6587
\GC78:79:-1:64% G4889
<0000e0000000000000000001f8000000000000000001fc000000000000000001ff000000%
 000000000003ff000000000000000003fe000000000000000003fe00000000003c000003%
 fc00000000007e000007fc0000000000ff000007f80381ffffffff800007f807c0ffffff%
 ffc0000ff00ff07fffffffc0000ff01ff8700fc00000fffffffffc000fc000007fffffff%
 fc000fc000003c1fc00000000fc00000001fc00000000fc00000001f800000000fc00000%
 003f800000000fc00000003f800000000fc00000003f800000000fc00000007f00000000%
 0fc00000007f700000000fc00000007f7c0000000fc00000007f7f0000000fc0000000fe%
 7fc000000fc0000000fe3fe000000fc0000000fc3fc000000fc0000001fc3fc000000fc0%
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 0fc0000007003e0000000fc0000002003e0000000fc0000000003e0000000fc000000000%
 3e0000000fc0000000003e0000000fc0000000003e0000000fc0000000003e0000000fc0%
 000000003e0000000fc0000000003e00fc000fc0000000003e07fc000fc0000000003e7f%
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 0fc000007fff3e0000000fc000003ffe3e0000000fc000001ff03e0000000fc000001fc0%
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 1fc0000000007f0000001fc0000000007f0000001fc000000000780000001e000000>%% u"8f69
), 348. % 2nd
Hoggatt, Verner Emil, Jr\period, 384. % 3rd
Hypergeometric functions, 205, 384. % 3rd
Incidence matrices, 122. % 3rd
Inorder traversal, 170. % 3rd
Inverse permutations, 27, 38--39, 146. % 3rd
Janson, Carl Svante, vi, 186, 194, 196, 200. % 3rd
Kajitani, Yoji (\JC48:48:0:40% J1965
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 >%% unicode char "68b6
\JC48:48:0:40% J3511
<0003000000000003800000000003f00f00000007f00780000007e003e000000fc000f800%
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 >%% unicode char "8c37
\JC48:48:0:40% J4546
<03800000020001e03e00030001f00f8007c000f807e007f0007c03f007e0007c01f80fc0%
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 >%% unicode char "6d0b
\JC45:48:0:40% J2742
<0000000000600000000000f0fffffffffff87ffffffffff80000000000f00000000000f0%
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 0030000000f00000000000f00000000000f00000000000f00000000000f00000000000f0%
 0000000000f00000000001f0000000003ff0000000000ff00000000007e00000000003c0%
 >%% unicode char "53f8
), 331. % 3rd
Kernels of a graph (maximal independent sets), 143, 244, 269. % 2nd
Kleiman, Mark Philip, 384. % 3rd
Knuth, Donald Ervin (\GC75:75:-3:62% G2463
<0000001f0000000000000000001ff8000000000000000007ff000000000000000003ff80%
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\GC73:74:-4:61% G2134
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\GC71:74:-3:61% G3641
<000000000003800000000000000003f00000000300000003fc00000003c0000003fc0000%
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), i, iv--vi, viii, 44, 52, 53, 61, 71, 75--77, 98, 116, 121, 193, 196, 200, 209, 214, 219, 221, 227, 228, 232--235, 237, 239--240, 248, 254, 267, 268, 271, 274, 281, 283--286, 288, 289, 292, 293, 309--311, 316, 319, 322, 331, 335, 340, 341, 346, 348, 349, 352, 356, 364, 368.
Komisarski, Andrzej, 183. % 3rd
L-twist, 80, 172, 336. % 3rd
Left-to-right maxima or minima, 384. % 3rd
Linear programming problems, 287, 332--333. % 2nd
Loops (cyclic paths), 141, 175--178, 345--346. % 3rd
Lou, Xingliang David (\GC79:78:-1:63% G3406
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\GC77:74:-2:62% G4839
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\GC79:80:0:64% G3333
<00000000380000000000000000007e0000000000000000003f0000000000000000001f80%
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 0000000000000000>%% unicode char "4eae
), 224. % 3rd
%Luria, Zur ({\heb\Haa/\Hyy/\Hrr/\Hvv/\Hll/ \Hrr/\Hvv/\Hts/}), 207. % 3rd
Magen, Avner ({\heb\Hfn/\Hgg/\Hmm/ \Hrr/\Hnn/\Hbb/\Haa/}), 26. % 3rd
{\mc MCC} problems: Multiple covering with colors, iv, 91--93, 121, 142--144, 239, 240, 251, 267--271, 280, 302, 303, 306, 325, 355. % 2nd
{\mc MCC} problems: Multiple covering with colors, iv, 91--93, 121, 142--144, 239, 240, 250, 251, 267--271, 280, 302, 303, 306, 325, 355. % 3rd
McDonald, Gary, 250, 356. % 2nd
Mebane, Palmer Croasdale, 346. % 3rd
Median function ($\langle xyz\rangle$), vi, 24, 201, 353. % 3rd
Median value of a random variable, 14, 204. % 3rd
Minimum cutsets, 344. % 2nd
Mirror images, \see Reflection symmetry. % 3rd
Modifications of Algorithm 7\period2\period2\period1C and related algorithms, 124, 125, 130, 131, 136, 181, 230, 250, 253, 350. % 2nd
Molinari, Rory Benedict, 344. % 2nd
Murata, Hiroshi (\JC48:48:0:40% J3428
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 003c00000f00003c00000f00003c0001ff00003c0000ff00003c00007e00001800003c00%
 >%% unicode char "6751
\JC41:46:-4:38% J3736
<c00000000600f00000000f00ffffffffff80ffffffffff80f0003c000f00f0003c000f00%
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\JC48:48:0:40% J4546
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 >%% unicode char "6d0b
), 331. % 3rd
$n$ queens problem (independent queens), 29--32, 44--46, 51--53, 68--71, 103, 108, 110--111, 116, 120, 121, 124--126, 143, 148, 150, 232. % 3rd
$n$-queens problem (dominating queens),  91--92, 142. % 2nd
Nixon, Dennison, 319. % 2nd
Nobel, Parth Talpur, 207. % 3rd
Odlyzko, Andrew Michael, 384. % 3rd
OEIS\regtm: The On-Line Encyclopedia of Integer Sequences\regtm\ ({\tt oeis\period org}), 230, 231, 263, 277, 312, 324. % 3rd
Ollerton, Richard Laurance, 384. % 3rd
Onnen, Hendrik, Sr\period, 32. % 2nd
Ordered options, \see Pairwise ordering trick. % 2nd
Pendant vertex: A vertex of degree one, 314. % 2nd
Permutations, 27, 32, 53, 101, 105, 122, 133, 135, 138, 146, 168, 351. % 3rd
Pi, as source of ``random'' data, 45--46, 48, 55, 58, 72, 76, 114--115, 126, 128, 144, 152, 158, 172--175, 180, 181, 346, 384.
Pinch, Ruth, 356. % 2nd
Pinter, Ron Yair ({\heb\Hrr/\Htt/\Hnn/\Hyy/\Hpp/  \Hrr/\Hyy/\Haa/\Hyy/ \Hfn/\Hvv/\Hrr/}), 331. % 3rd
Pittel, Boris Gershon ({\rus Pittelp1, Boris Gershonovich}), 196, 231. % 3rd
Planar graphs, 112, 344. % 3rd
Projection vectors, 345, 346--348. % 2nd
Puzzle Square JP, 346. % 3rd
Queen domination problems, 91--92, 142. % 2nd
R-twist, \see L-twist. % 3rd
Recurrence relations, 99--101, 146, 192, 197, 216, 226, 355, 384. % 3rd
Reflection symmetry, 40, 53, 81, 89, 165, 169, 172, 206, 236--237, 257, 260--262, 303. % 3rd
Right-to-left maxima or minima, 384. % 3rd
Rooms and bounds, 169--170, 384.
{\mc SAT} solvers, 148, 213--214, 235, 274, 296, 303, 321, 346. % 3rd
Simkin, Menahem Michael\indexbreak ({\heb\Hfn/\Hvv/\Hqq/\Hmm/\Hvv/\Hss/ \Hll/\Haa/\Hcc/\Hyy/\Hmm/ \Hfm/\Hhh/\Hnn/\Hmm/}), 53. % 3rd
Social distancing, 157. % 3rd
Spanning trees, 60, 353. % 3rd
Stappers, Filip Jan Jos, 234. % 2nd
Stork, David Geoffrey, 183. % 2nd
Tatami tilings, 155, 169--170, 240, 296, 305, 328. % 3rd
Ternary commafree codes, 36--37, 39, 214. % 3rd
Torczon, Linda Marie, 39. % 3rd
Tree insertion, 330. % 3rd
Trick shapes, 162. % 3rd
Trybu{\l}a, Stanis{\l}aw Czes{\l}aw, 183. % 2nd
Twin tree structure, 170. % 3rd
VLSI layout, 331. % 3rd
Watanabe, Tomomi (\JC48:48:0:40% J3747
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 0fc3803eff800fc3807c7ffe0fc600f03ffe0fc603c01ffc07cc1f0007fc0788f80001f8%
 >%% unicode char "6e21
\JC48:45:0:37% J4253
<0000000000303800000000781f01fffffffc0fc0fffffffc07e0007c007803f0007c0078%
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\JC48:47:0:39% J3546
<006000000000007c00000000007c0000000000780000000000780000000000f0000c001c%
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 003c3c0f003c00781e0f003c00781e0f003c00f00f0ffffc00f00f0ffffc01e0070f003c%
 03c0070f003c0780000f003c0f00000600183c0000000000f00000000000>%% uni "77e5
\JC44:46:-4:38% J2442
<000000001c00000000003e00ffffffffff007fffffffff00000000003c00000000003c00%
 000000003c00000000003c00000000003c00000000003c00000000003c00000000003c00%
 000000003c00000000003c00c00000003c00e00000003c00fffffffffc00fffffffffc00%
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 f000000000e0f000000000f0f000000000f0f000000000f0f800000001f0fc00000003f0%
 fffffffffff0fffffffffff07fffffffffe03fffffffffc0>%% unicode "5df1
), 331. % 3rd
Windsor, Aaron Andrew, 213, 214, 250. % 3rd
\.{WORDS}($n$), the $n$ most common five-letter words of English, 34, 54, 60, 92, 131, 181, 209, 242, 251, 271--272, 292. % 2nd
Yano, Tatsuo (= Ryouh, \JC48:47:0:40% J4480
<000800000000000e00000000000f80000000000fe0000000001ff0000000001fe0000000%
 001fc0000000001f80000000003f80000600003f00000f00003fffffff80007fffffff80%
 007c0600000000f807c0000000f007e0000001e007e0000003c007c00000038007c00000%
 070007c000000e0007c00000180007c00000300007c00018000007c0003c7ffffffffffe%
 3ffffffffffe000007c00000000007c0000000000fc0000000000fe0000000001fe00000%
 00001f70000000003e70000000003e38000000007e38000000007c1c00000000f81e0000%
 0001f80f00000003f00780000007e007c000000fc003f000001f8001fc00003f0000ff80%
 00fe00007ff801f800003fff07e000000ffe3f00000003fef8000000007c>%% unicode "77e2
\JC48:48:0:40% J4478
<3000180000183c003c00003c3ffffefffffe3ffffe7ffffe3c3c3c0001fc3c3c3c0001f8%
 3c3c3c0003f03c3c3c0003e03c3c3c0007803c3c3c1c07003c3c3c070c003c3c3c07c000%
 3ffffc03e0003ffffc01f0003c3c3c01f0003c3c3c00f0003c3c3c00e00c3c3c3c00001e%
 3c3c3fffffff3c3c3dffffff3c3c3c00f83f3c3c3c00f83e3ffffc00f87c3ffffc00f878%
 3c3c3c00f8f0183c1800f8e0003c0000f9c0003c0000f980003c0000fb00003c1800fa00%
 003c3c00f8003ffffe00f8001ffffe00f800003c0000f800003c0000f800003c0000f800%
 003c0000f800003c0000f800003c0000f800003c3fc0f800007ffc00f800fffff000f800%
 ffff0000f8007ff80001f8007fe0003ff8007c000007f80020000001f00000000000e000%
 >%% unicode char "91ce
\JC48:48:0:40% J4622
<003000600000003c00780000003e007c0000003e007c0060003c307800f0003c787ffff8%
 3ffffc7ffff81ffffc7800000303807800000383c07800c001c3e07801e001c3c07ffff0%
 01e7807ffff001e7007801e001e70c3001e000c61e0001e0ffffff0001e07fffff6001e0%
 0000007801e00c00607fffe00f00f07fffe00ffff87801e00ffff87800c00f00f0780000%
 0f00f07803000f00f07807800f00f07fffc00ffff07fffc00ffff07800000f00f0780000%
 0f00f07803000f00f07807800f00f07fffc00ffff07fffc00ffff07800000f00f0780000%
 0f00f07803000f00f07807840f00f07fffc40f00f07fffc40f00f078000c0f00f078000e%
 0f00f078000e0f01f078001f0f3ff07c003f0f07f07fffff0f01f07ffffe0600e03ffffc%
 >%% unicode char "9f8d
\JC48:44:0:38% J1806
<0000000000300000000000783ffffffffffc1ffffffffffc000003c00000000003c00000%
 000003c00000000003c00000000003c00000000003c00000000003c00000000003c00000%
 000003c00000000003c00000000003c00000000003c00000000003c00000000003c00000%
 000003c00000000003c000c0000003c001e00ffffffffff007fffffffff0000003c00000%
 000003c00000000003c00000000003c00000000003c00000000003c00000000003c00000%
 000003c00000000003c00000000003c00000000003c00000000003c00000000003c00000%
 000003c00000000003c00000000003c00000000003c00000000003c0001c000003c0003e%
 ffffffffffff7fffffffffff>%% unicode char "738b
), 340, 346. % 3rd
Yao, Bo (\GC77:78:-1:63% G5006
<0003c0000000000000000003f000000003c000000003fc00001c03e000000003fc00001f%
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 001f83f00000000ff000001f83f00000000ff070001f83f00000000ff0f8001f83f00000%
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 003e03f0001f83f00000007e03f0001f83f00000007e03f0001f83f00000007c03f0001f%
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 07e003df83f1fe0001f807c0079f83f0ff0001f80fc01f1f83f07f8001f00fc7fe1f83f0%
 7f8003f01f87fe1f83f03fc003f01f83fc1f03f01fc003e03f01f81f03f01fc001f83f00%
 f03f03f00fc000fe7e00603f03f007c0003ffe00003f03f003c0000ffc00003f03f00100%
 0007fc00007f03f000000003fc00007e03f000000001ff0000fe03f000000001ff8000fc%
 03f000c00003ffe001fc03f000c00003eff001fc03f000e00007c7fc03f803f000e0000f%
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 00000000000007c000000000000000000200000000000000>%% unicode char "59da
\GC76:77:-3:63% G1808
<00000000000700000000000000000007c0000000000000000007f0000000000000000007%
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 002fc01ff8000003ffc0001f00ffe0000000ff00001c0fff000000003c0000100ffc0000%
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), 331. % 3rd

\vfill % TEMPORARY

\enddoublecolumns
\endgroup
\endchange

\vfill\eject
\amendpage 4f5.384 a new section following the index (21.09.18)
\bigskip
\rightline{\titlefont BREAKING NEWS}
\bigskip\noindent
\tenpoint
\pageref|breaking|{\tencsc Here}
 are answers to \MPR\ exercises that were recently added.
\bigskip
\ninepoint
\ans135.
 (Notice that the smallest {\it non\/}-Baxter permutations
are 3142 and its inverse, 2413.)\par
If $P$ is a Baxter permutation, so are $P^R=p_n\ldots p_1$ and
$P^C=\bar p_1\ldots \bar p_n$, where $\bar x=n+1-x$.
So is the permutation $P\setminus n$ obtained by
deleting~$n$; and so are the permutations $P\setminus x$ obtained by
deleting~$x$ and subtracting 1 from each element that exceeds~$x$, if
$x=p_n$ or $x=1$ or $x=p_1$. (Consider, for example, deleting~$n$ from $P^-$.)
\par\breathe
 Let's look at the $n+1$ permutations obtained by {\it inserting\/} $n+1$
into an $n$-element Baxter permutation. For example, when $n=8$ and
$P=21836745$ the nine extensions are
921836745,
291836745,
219836745,
218936745,
218396745,
218369745,
218367945,
218367495,
218367459.
Only four of these fail Baxter's property, namely
291836745, 218396745, 218369745, and 218367495; and we soon discover the
general rule: {\sl $n+1$ can be Baxterly inserted if and only if it's placed
just before a left-to-right maximum, or just after a right-to-left
maximum.} (In our example, the left-to-right maxima are 2 and~8; the
right-to-left maxima are 5, 7, and~8.)\par
Let $B_n(i,j,k)$ be the number of $(n+1)$-element Baxter permutations
with exactly $i+1$ left-to-right maxima, $j+1$ left-to-right minima,
$k$~ascents, and $n-k$~descents.
Such permutations correspond to floorplans with
$n+1$ rooms, $i+1$ rooms touching the bottom of the frame,
$j+1$ rooms touching the left of the frame, $k+2$ vertical bounds,
and $n-k+2$ horizontal bounds (see exercise 7.2.2.1--372).
The reasoning above yields the interesting recurrence
$$\medmuskip=1mu
B_1(i,j,k)=\[i=j=k=0],\
B_{n+1}(i+1,j+1,k)=\sum_{i'>i}B_n(i',j,k)\,+\sum_{j'>j}B_n(i,j',k-1);$$
% B_n(i,j,k)=B_n(j,i,n-k)
and the solution can be expressed as a determinant of
binomial coefficients:
$$B_n(i,j,k)=\det\pmatrix{
{n-j-1\choose k-1} & {n\choose k-1} & {n-i-1\choose n-k+1}\cr
{n-j-1\choose k}   & {n\choose k}   & {n-i-1\choose n-k}\cr
{n-j-1\choose k+1} & {n\choose k+1} & {n-i-1\choose n-k-1}\cr
},\ \hbox{unless $\left\{\vcenter{\halign{#\hfil\cr
                             $i=0$ and $j=n$\cr
                             \quad or\cr
                             $i=n$ and $j=0$.\cr}}\right.$}$$
% exceptions are cases where {x\choose-1} is sensitive...
Summing on $i$ and $j$ now gives the simpler formula
$$b_n(k)=t_{n+1}(k+1)/t_{n+1}(1),\quad\hbox{where
  $t_n(k)={n\choose k-1}{n\choose k}{n\choose k+1}$,}$$
for the number of $n$-element Baxter permutations with exactly $k$ ascents.\par
Since the terms with $k\approx n/2$ dominate the sum
$b_n=\sum_k b_n(k)$, we obtain the asymptotic value
$$b_n={8^{n+2}\over\sqrt{12} \pi n^4}\Bigl(1-{22\over3n}+O(n^{-2})\Bigr),$$
due to A.~M. Odlyzko.
[See G. Baxter, {\sl Proc.\ American Math.\ Soc.\ \bf15} (1964), 851--855;
F.~R.~K. Chung, R.~L. Graham, V.~E. Hoggatt,~Jr., and
M.~Kleiman, {\sl Journal of Combinatorial Theory\/ \bf A24} (1978),
382--394;
W.~M. Boyce, {\sl Houston J. Math. \bf7} (1981), 175--189;
S.~Dulucq and O.~Guibert, {\sl Discrete Math.\ \bf180} (1998),
143--156.] R.~L. Ollerton has found the recurrence
$(n+2)(n+3)b_n=(7n^2+7n-2)b_{n-1}+8(n-\nobreak1)(n-2)b_{n-2}$, with $b_1=1$,
as well as the closed form
$b_n=F\bigl({1-n,-n,-1-n\atop2,3}\bigm\vert-1\bigr)$.
The initial terms are $(b_0,b_1,\ldots{})=(1$, 1, 2, 6, 22, 92, 422,
2074, 10754, 58202, \dots). % OEIS A1181
\endchange

\amendpage 4f5.384 new answer (21.09.26)

\ans136. It's true if $y\le x+\half$, because $f(x+t)-f(x)$ increases from
$f(t)$ to $-f(1-t)$ as $x$ increases from 0 to~$1-t$.
But it fails when $x<\half$ and $y=1$.
\endchange

\bye

[The next printing will be the 3rd.]

not listed above:
page 30, I've made the lines of Fig 68 twice as thick
page 45, lines 11-12 after the caption, "Incidentally, ..."
page 56, better wording of exercise 42
page 65, changed wording just before (9)
page 85, line 5, "appears" -> "can be found"
page 101, line -6, better spacing
page 177, line 2 of exercise 423, boolean -> Boolean
page 185 gains a line because of the change to page 186
page 210, ans 32: A K Peters
page 219, better line breaks in answer 70
page 227, line -2 of answer 21, "; for" -> ". For"
page 242, line -2 of answer 86, "Sum those estimates"
page 259, line 6, "is a" -> "are a"
page 268, ans 154: A K Peters
page 288, line -4, fix the spacing
page 309, line 1 of ans 313, Pieces -> pieces
page 321, line -10, `#' -> #
page 346, 2nd line of answer 419, more-or-less
and I've made the polyspheres a bit darker (pages 167, 324)
delete the following from the index (because of the change to ex 7.2.2.1--372):
 Biggs, Norman Linstead, 331.
 Digraphs, orthogonal, 169--170.
 Dual of a planar graph, 331.
 Laplace (= de la Place), Pierre Simon, Marquis de, matrix, 331.
 Orthogonal digraphs, 169--170.
 Sphere, 170.
 Stereographic projection, 170.
 Transitive tournaments, 170.
also delete Zur Loria (because of the change to ex 7.2.2--15)

ARTICLES "TO APPEAR" THAT ARE STILL PENDING:
Fischetti, Salvagnin, ans 7.2.2.1--36 (arXiv:1907.08246 [cs.DS])
Harris's G4G9 presentation, ans 7.2.2.1--62
Beluhov snake-in-box knight paths, ans 7.2.2.1--173
Beluhov symmetric plane partitions and diamond/lozenge tilings, ans 7.2.2.1--261
Svante Janson arXiv:2009.13781 [ans MPR24(c)]
Windsor's commafree codes? (I asked him Dec21 to tell me if he publishes)
Simkin arXiv 2107.13460 [ans 7.2.2--15]