Conjectures about set partitions and generating functions
Let a_{nk} be the number of partitions of {1,...n} whose "final" part
has size k. For instance, here are the partitions when n=4:
1234 [1111] 4
123|4 [1112] 1
124|3 [1121] 1
12|34 [1122] 2
12|3|4 [1123] 1
134|2 [1211] 1
13|24 [1212] 2
13|2|4 [1213] 1
14|23 [1221] 2
1|234 [1222] 3
1|23|4 [1223] 1
14|2|3 [1231] 1
1|24|3 [1232] 1
1|2|34 [1233] 2
1|2|3|4 [1234] 1
With each partition I've shown its restricted growth sequence, and the
size of its final part (which is the number of occurrences of the
maximum digit of its restricted growth sequence). Thus we have
a_{41}=9, a_{42}=4, a_{43}=1, a_{44}=1.
Tabulating these numbers and looking at OEIS [see A121207] leads us to realize
that the matrix (a_{nk}), extended so that the subscripts run from 0
instead of from 1, is the inverse of the triangular matrix that begins
1 0 0 0 0 0
-1 1 0 0 0 0
-1 -1 1 0 0 0
-1 -2 -1 1 0 0
-1 -3 -3 -1 1 0
-1 -4 -6 -4 -1 1
etc., namely the negative of Pascal's triangle, below all 1s on the diagonal.
In this extended form the first column contains the numbers a_{n0} for
n=0, 1, ..., which clearly are the Bell numbers $\varpi_n$:
a_{n0} = a_{n1} + a_{n2} + ... a_{nn}
and a_{40}=\varpi_4=15.
All of this is easy to prove, extending a remark found in OEIS A040027.
Now let n become infinite, and consider the limiting value of
a_{nk}/a_{n0} for fixed k. Call it \alpha_k. This is, of course,
a probability distribution, on the size of the final part.
Empirically, convergence is very fast, leading to:
\alpha_1 = .59634736232319
\alpha_2 = .26596538503241
\alpha_3 = .09678032513886
And here we recognize that alpha_1 is the Euler--Gompertz constant!
For fixed k, let A_k(z) be the exponential generating function
\sum_n a_{nk} z^n/n!.
Then A_0(z) is the well known generating function for Bell numbers,
e^{e^z-1}; call it B(z). One can also prove that
A_{k+1}'(z) = z^k/k! + e^z A_{k+1}(z)
hence
A_{k+1}(z) = B(z) \int_0^z t^k/B(t) dt / k!.
I showed this to Ira Gessel, who discovered empirically that
\alpha_{k+1} = \int_0^\infty t^k/B(t) dt / k!.
More generally, he defined
T_{r,s}(z) = B(z) \int_0^z t^r e^{st}/B(t) dt
and made the empirical conjecture
[z^n] T_{r,s}(z)
---------------- --> \int_0^\infty t^r e^{st}/B(t) dt.
[z^n] B(z)
There is strong numerical evidence for this conjecture;
"it must be true."
That leads to the question, for what power series G(z) are
the coefficients of B(z)G(z) asymptotically proportional
to G(\infty) times the coefficients of B(z)??
-- Don Knuth, Pite{\aa} Sweden, 6 January 2018
P.S. Gessel also pointed out that the traditional formula
for the Euler--Gompertz constant, \int_0^\infty e^{-x} dx/(1+x),
follows by the substitution t=\ln(1+x). We can also write this
as \int_0^\infty e^{-x}\ln(1+x) dx, if my hand calculations
can be trusted. More generally, the conjectured formula
above translates into
\alpha_k = \int_0^\infty e^{-x} (\ln(1+x))^k dx / k!.
P.P.S. News flash from Gessel, 7 January: The conjectured
values of \alpha_k may have been proved by Asakly, Blecher,
Brennan, Knopfmacher, Mansour, and Wagner, in J Math
Anal Appl 416 (2014) 672--682!