Dominique Foata

A Combinatorial Proof of Bass’s Evaluations of the Ihara-Selberg Zeta Function for Graphs

We derive combinatorial proofs of the main two evaluations of the Ihara-Selberg Zeta function associated with a graph. We give three proofs of the first evaluation all based on the algebra of Lyndon words. In the third proof it is shown that the first evaluation is an immediate consequence of Amitsurs identity on the characteristic polynomial of a sum of matrices. The second evaluation of the Ihara-Selberg Zeta function is first derived by means of a sign-changing involution technique. Our second approach makes use of a short matrix-algebra argument.

Mappings of acyclic and parking functions

The two functions in question are mappings: [n]→[n], with [n] = {1, 2,⋯,n}. The acyclic function may be represented by forests of labeled rooted trees, or by free trees withn + 1 points; the parking functions are associated with the simplest ballot problem. The total number of each is (n + 1)n-1. The first of two mappings given is based on a simple mapping, due to H. O. Pollak, of parking functions on tree codes. In the second, each kind of function is mapped on permutations, arising naturally from characterizations of the functions. Several enumerations are given to indicate uses of the mappings.

A combinatorial proof of the Mehler formula

Abstract A combinatorial proof of the Mehler formula on Hermite polynomials is given that is based upon the techniques of the partitional complex.

Laguerre polynomials, weighted derangements, and positivity

A calculation of the linearization coefficients of the (generalized) Laguerre polynomials

Rearrangements of the symmetric group and enumerative properties of the tangent and secant numbers

L_n^{( \alpha )} ( x )

Distributions Euleriennes et Mahoniennes sur le Groupe des Permutations

is proposed by means of analytic and combinatorial methods. This paper extends to the case of an arbitrary

Eulerian Polynomials: From Euler’s Time to the Present

\alpha

A Classic Proof of a Recurrence for a Very Classical Sequence

a combinatoric and analytic result due to Askey, Ismail, and Koornwinder and Even and Gillis.

Eulerian calculus, II: an extension of Han's fundamental transformation

In a recent note [6] the first author has announced the discovery of a family of transformation groups (G,), > o which have the following property: G, acts on the n! elements of the symmetric group ~ , and the number of its orbits is equal to the n-th tangent or secant number, according as n is odd or even. The purpose of this paper is to give a complete description of these groups. Applications to enumeration problems will appear in a subsequent paper. The tangent (or Euler) number are defined by the series expansion of tan u

Congruences for the q-secant Numbers

Les nombres euleriens An,k (n ≥ 1, 1 ≤ k ≤ n) sont definis par la relation de recurrence