Reference request: Gessel interview’s generating function identities

algebraic-combinatoricsco.combinatoricscombinatorial-identitiesgenerating-functionsgraph theory

In this interview, Ira Gessel mentions the following results:

Result 1: Let $B_n$ denote the $n^{\text{th}}$ Bernoulli number.
Define the series
$$B(x) = \sum_{n=2}^{\infty} \frac{B_nx^{n-1}}{n(n-1)}.$$
Let $v_n$ be the sequence of reals with
such that the series
$$F(x) = \sum_{n=1}^{\infty} \frac{v_nx^n}{n!}$$
$$F\left(\sqrt{2x – 2(\log(1+x))}\right) = x.$$
Then we have
$$e^{B(x)} = \sum_{n=0}^{\infty} \frac{v_{2n+1}x^n}{2^n n!}.$$

Result 2: Let $s_n$ be the number of strongly connected, directed graphs with vertex set $\{1, \dots, n\}$. Let $t_n$ be the number of strongly connected tournaments with vertex set $\{1, \dots, n\}$. Let
$$T(x) = \sum_{n=1}^{\infty} \frac{2^{\binom{n}{2}}t_n x^n}{n!}.$$
$$\log\left(\frac{1}{1-T(x)}\right) = \sum_{n=1}^{\infty} \frac{s_nx^n}{n!}.$$

Result 3: Let $A_n(t)$ denote the $n^{\text{th}}$ Eulerian polynomial. Then
$$\frac{A_n(t)}{(1-t)^{n+1}} = \sum_{k=0}^{\infty} k^nt^k.$$

My Question: What are the references for the proofs of results 1 through 3 above?

Best Answer

Result 1 can be found in N. G. de Bruijn, Asymptotic Methods in Analysis, Dover, New York, 1981, p. 71.

Result 2 follows from a formula of E. M. Wright, though he didn't state it in this form. A discussion of this formula, with a generalization, can be found in Kassie Archer, Ira M. Gessel, Christina Graves, and Xuming Liang, Counting acyclic and strong digraphs by descents, Discrete Math. 343 (2020), 112041, 14 pp. See Proposition 8. The arXiv version is

As Sam and Richard noted, Result 3 is well known. It is equivalent to Euler's definition of the Eulerian polynomials. The combinatorial interpretation of the Eulerian polynomials is a special case of a much more general result of MacMahon, though I don't think that MacMahon recognized this special case as being noteworthy, nor did he connect it with Euler's work.

Related Question