Ira M. Gessel

Brandeis University · Department of Mathematics

The ring of cyclic quasi-symmetric functions and its non-Escher subring are introduced in this paper. A natural basis consists of fundamental cyclic quasi-symmetric functions; for the non-Escher subring they arise as toric P-partition enumerators, for toric posets P with a total cyclic order. The associated structure constants are determined by cyc...

We give a short proof, using generating functions, for a polynomial congruence for Eulerian polynomials first proved, using arrangements of hyperplanes, by Yoshinaga and later proved, using roots of unity, by Iijima, Sasaki, Takahashi, and Yoshinaga.

We prove several general formulas for the distributions of various permutation statistics over any set of permutations whose quasisymmetric generating function is a symmetric function. Our formulas involve certain kinds of plethystic substitutions on quasisymmetric generating functions, and the permutation statistics we consider include the descent...

Let A be an alphabet and let F be a set of words with letters in A. We show that the sum of all words with letters in A with no consecutive subwords in F, as a formal power series in noncommuting variables, is the reciprocal of a series with all coefficients 0, 1 or -1. We also explain how this result is related to a result of Curtis Greene on latt...

A descent of a labeled digraph is a directed edge (s,t) with s>t. We count strong tournaments, strong digraphs, acyclic digraphs, and forests by descents and edges. To count strong tournaments we use Eulerian generating functions and to count strong and acyclic digraphs we use a new type of generating function that we call a graphic Eulerian genera...

In this note we generalize an identity of John Riordan and Robert Donaghey relating the enumerator for Stirling permutations to the Eulerian polynomials.

We prove several general formulas for the distributions of various permutation statistics over any set of permutations whose quasisymmetric generating function is a symmetric function. Our formulas involve certain kinds of plethystic substitutions on quasisymmetric generating functions, and the permutation statistics we consider include the descent...

In 1995, the first author introduced a multivariate generating function G that tracks the distribution of ascents and descents in labeled binary trees. In addition to proving that G is symmetric, he conjectured that G is Schur positive. We prove this conjecture by expanding G positively in terms of ribbon Schur functions. We obtain this expansion u...

A descent of a labeled digraph is a directed edge (s, t) with s > t. We count strong tournaments, strong digraphs, and acyclic digraphs by descents and edges. To count strong tournaments we use Eulerian generating functions and to count strong and acyclic digraphs we use a new type of generating function that we call a graphic Eulerian generating f...

The ring of cyclic quasi-symmetric functions and its non-Escher subring are introduced in this paper. A natural basis consists of fundamental cyclic quasi-symmetric functions; for the non-Escher subring they arise as toric $P$-partition enumerators, for toric posets $P$ with a total cyclic order. The associated structure constants are determined by...

We show that the reciprocal of a partial sum with 2m terms of the alternating exponential series is the exponential generating function for permutations in which every increasing run has length congruent to 0 or 1 modulo 2m. More generally we study polynomials whose reciprocals are exponential generating functions for permutations whose run lengths...

Egge, Loehr, and Warrington proved a formula for the Schur function expansion of a symmetric function in terms of its expansion in fundamental quasi-symmetric functions. Their formula involves the coefficients of a modified inverse Kostka matrix. Recently Garsia and Remmel gave a simpler reformulation of Egge, Loehr, and Warrington's result, with a...

In 1995, Gessel introduced a multivariate generating function that tracks the distribution of ascents and descents in labeled binary trees. In addition to proving that it is symmetric, he conjectured that it is Schur-positive. We prove this conjecture by developing a weighted extension of a bijection of Pr\'eville-Ratelle and Viennot relating pairs...

We introduce the notion of a shuffle-compatible permutation statistic, which captures a property of several descent statistics previously observed by Richard Stanley, and the shuffle algebra associated to a shuffle-compatible permutation statistic. This paper develops a theory of shuffle-compatibility for descent statistics. We prove a shuffle-comp...

We give a survey of the Lagrange inversion formula, including different versions and proofs, with applications to combinatorial and formal power series identities.

A tanglegram is a diagram, used in biology to compare phylogenetic trees, consisting of two binary trees together with a matching of their leaves. Unlabeled tanglegrams were recently counted by Billey, Konvalinka, and Matsen. Using the theory of combinatorial species, we count three variations of unlabeled tanglegrams: unordered, unrooted, and unor...

We give a historical survey of the theory P-partitions, starting with MacMahon's work, describing Richard Stanley's contributions and his related work, and continuing with more recent developments.

We give a simple proof of George Andrews’s balanced 5 F 4 evaluation using two fundamental principles: the nth difference of a polynomial of degree less than n is zero, and a polynomial of degree n that vanishes at n+1 points is identically zero.

We present an empirical-yet-rigorous approach for solving a wide class of functional equations, thereby automating many results that previously required considerable human ingenuity and human labor.

We find the exponential generating function for permutations with all valleys even and all peaks odd, and use it to determine the asymptotics for its coefficients, answering a question posed by Liviu Nicolaescu. The generating function can be expressed as the reciprocal of a sum involving Euler numbers. We give two proofs of the formula. The first...

We give a short proof, using Lagrange inversion, of a congruence modulo 3 for the number of connected noncrossing graphs on n vertices that was conjectured by Emeric Deutsch and Bruce Sagan. A more complicated proof had been given earlier by S.-P. Eu, S.-C. Liu, and Y.-N. Yeh.

We investigate constraints on embeddings of a non-orientable surface in a $4$-manifold with the homology of $M \times I$, where $M$ is a rational homology $3$-sphere. The constraints take the form of inequalities involving the genus and normal Euler class of the surface, and either the Ozsv\'ath--Sazb\'o $d$-invariants or Atiyah--Singer $\rho$-inva...

We count unlabeled k-trees by properly coloring them in k+1 colors and then counting orbits of these colorings under the action of the symmetric group on the colors.

We use the theory of combinatorial species to count unlabelled bipartite graphs and bipartite blocks (nonseparable or 2-connected graphs). We start with bicolored graphs, which are bipartite graphs that are properly colored in two colors. The two-element group S2 acts on these graphs by switching the colors, and connected bipartite graphs are orbit...

We give a simple proof of George Andrews's balanced 5F4 evaluation using two fundamental principles: the nth difference of a polynomial of degree less than n is zero, and a polynomial of degree n that vanishes at n+1 points is identically zero.

We study formulas expressing Fibonacci numbers as sums over compositions using free submonoids of the free monoid of compositions with parts 1 and 2.

We investigate the diagonal generating function of the Jacobi–Stirling numbers of the second kind JS(n+k,n;z) by generalizing the analogous results for the Stirling and Legendre–Stirling numbers. More precisely, letting JS(n+k,n;z)=pk,0(n)+pk,1(n)z+⋯+pk,k(n)zk, we show that (1−t)3k−i+1∑n≥0pk,i(n)tn(1−t)3k−i+1∑n≥0pk,i(n)tn is a polynomial in tt with...

Two of the present authors have given in 1993 a bijection ΦΦ between words on a totally ordered alphabet and multisets of primitive necklaces. At the same time and independently, Burrows and Wheeler gave a data compression algorithm which turns out to be a particular case of the inverse of ΦΦ. In the present article, we show that if one replaces in...

We prove a conjecture of Drake and Kim: the number of 2-distant noncrossing partitions of {1,2,…,n} is equal to the sum of weights of Motzkin paths of length n, where the weight of a Motzkin path is a product of certain fractions involving Fibonacci numbers. We provide two proofs of their conjecture: one uses continued fractions and the other is co...

Charlier configurations provide a combinatorial model for Charlier polynomials. We use this model to give a combinatorial proof of a multilinear generating function for Charlier polynomials. As special cases of the multilinear generating function, we obtain the bilinear generating function for Charlier polynomials and formulas for derangements.

Let p be a prime. The Hilbert-Kunz multiplicity, mu, of the element sum(x_i^(d_i)) of (Z/p)[x_1,..., x_s] depends on p in a complicated way. We calculate the limit of mu as p -> infinity. In particular when each d_i is 2 we show that the limit is 1 + the coefficient of z^(s-1) in the power series expansion of sec z + tan z. Comment: 6 pages

We prove a conjecture of Drake and Kim on a continued fraction. Comment: 6 pages, 2 figures

Carlitz, Handa, and Mohanty proved determinantal formulas for counting partitions contained in a fixed bounding shape by area. Gessel and Viennot introduced a combinatorial method for proving such formulas by interpreting the determinants as counting suitable configurations of signed lattice paths. This note describes an alternative combinatorial a...

We present a third proof of an interesting binomial coefficient identity using a generating function approach.

We present a third proof of an interesting binomial coefficient identity using a generating function approach.

Given integers $a_1, a_2, ..., a_n$, with $a_1 + a_2 + ... + a_n \geq 1$, a symmetrically constrained composition $\lambda_1 + lambda_2 + ... + lambda_n = M$ of $M$ into $n$ nonnegative parts is one that satisfies each of the the $n!$ constraints ${\sum_{i=1}^n a_i \lambda_{\pi(i)} \geq 0 : \pi \in S_n}$. We show how to compute the generating funct...

We compute the joint distribution of descent and major index over permutations of {1,..., n} with no descents in positions {n−i, n−i+1, ... , n−1} for fixed i≥ 0. This was motivated by the problem of enumerating symmetrically constrained compositions and generalizes Carlitz’s q-Eulerian polynomial.

Point-determining graphs are graphs in which no two vertices have the same neighborhoods, co-point-determining graphs are those whose complements are point-determining, and bi-point-determining graphs are those both point-determining and co-point-determining. Bicolored point-determining graphs are point-determining graphs whose vertices are properl...

We prove and generalize a conjecture of Goulden, Litsyn, and Shevelev that certain Laurent polynomials related to the solution of a functional equation have only odd negative powers.

Adin, Brenti, and Roichman [R.M. Adin, F. Brenti, Y. Roichman, Descent numbers and major indices for the hyperoctahedral group, Adv. in Appl. Math. 27 (2001) 210-224], in answering a question posed by Foata, introduced two descent numbers and major indices for the hyperoctahedral group , whose joint distribution generalizes an identity due to MacMa...

We introduce a statistic pmaj(P) for partitions of [n], and show that it is equidistributed with cr2, the number of 2-crossings, over all partitions of [n] with given sets of minimal block elements and maximal block elements. This generalizes the classical result of equidistribution for the permutation statistics inv and maj.

We introduce an elementary method to give unified proofs of the Dyson, Morris, and Aomoto identities for constant terms of Laurent polynomials. These identities can be expressed as equalities of polynomials and thus can be proved by verifying them for sufficiently many values, usually at negative integers where they vanish. Our method also proves s...

This tutorial talk will describe some of the basic problems and methods of lattice path enumeration, including the reflection principle, the method of images, the cycle lemma, generating functions, and the kernel method.

Without Abstract

I. P. Goulden, S. Litsyn, and V. Shevelev [On a sequence arising in algebraic geometry, J. Integer Sequences 8 (2005), 05.4.7] conjectured that certain Laurent polynomials associated with the solution of a functional equation have only odd negative powers. We prove their conjecture and generalize it.

A q-analog of functional composition for Eulerian generating functions is introduced and applied to the enumeration of permutations by inversions and distribution of left-right maxima.

Thesis. 1977. Ph.D.--Massachusetts Institute of Technology. Dept. of Mathematics. MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE. Vita. Bibliography : leaves 104-110. Ph.D.

A proper vertex of a rooted tree with totally ordered vertices is a vertex that is less than all its proper descendants. We count several kinds of labeled rooted trees and forests by the number of proper vertices. Our results are all expressed in terms of the polynomials P_n(a,b,c)= c(a+(n-1)b+c)(2a+(n-2)b+c)...((n-1)a+b+c) which reduce to (n+1)^{n...

One form of the inclusion-exclusion principle asserts that if A and B are functions of finite sets then A(S) is the sum of B(T) over all subsets T of S if and only if B(S) is the sum of (-1)^|S-T| A(T) over all subsets T of S. If we replace B(S) with (-1)^|S| B(S), we get a symmetric form of inclusion-exclusion: A(S) is the sum of (-1)^|T| B(T) ove...

Michael Somos conjectured a relation between Hankel determinants whose entries $\frac 1{2n+1}\binom{3n}n$ count ternary trees and the number of certain plane partitions and alternating sign matrices. Tamm evaluated these determinants by showing that the generating function for these entries has a continued fraction that is a special case of Gauss's...

It is well known that the numbers (2m)!(2n)!/m!n!(m+n)! are integers, but in general there is no known combinatorial interpretation for them. When m=0 these numbers are the middle binomial coefficients C(2n,n), and when m=1 they are twice the Catalan numbers. In this paper, we give combinatorial interpretations for these numbers when m=2 or 3.

We give a short proof of Miki's identity for Bernoulli numbers, n-2 # i=2 # i #n-i - n-2 # i=2 # n i # # i #n-i =2Hn#n , for n # 4 where, # i = B i /i, B i is the ith Bernoulli number, and Hn =1+1/2+ +1/n. 1. Introduction. The Bernoulli numbers B n are defined by # # n=0 B n x n n! = x e x - 1 . There are many known convolution identities for Berno...

A classical result of MacMahon gives a simple product formula for the generating function of major index over the symmetric group. A similar factorial-type product formula for the generating function of major index together with sign was given by Gessel and Simion. Several extensions are given in this paper, including a recurrence formula, a specia...

We give a formal Laurent series proof of Andrews's $q$-Dyson Conjecture, first proved by Zeilberger and Bressoud.

Let Phi(u, v) = Sigma(m=0)(infinity) Sigma(n=0)(infinity) c(mn) u(m) v(m). Bouwkamp and de Bruijn found that there exists a power series Psi(u, v) satisfying the equation t Psi(tz, z) = log(Sigma(k=0)(infinity) t(k)/k! exp(k Phi(kz, z))). We show that this result can be interpreted combinatorially using hypergraphs. We also explain some facts about...

Some of the classical orthogonal polynomials such as Hermite, Laguerre, Charlier, etc. have been shown to be the generating polynomials for certain combinatorial objects. These combinatorial interpretations are used to prove new identities and generating functions involving these polynomials. In this paper we apply Foata's combinatorial interpretat...

It is well known that the numbers $(2m)! (2n)!/m! n! (m+n)!$ are integers, but in general there is no known combinatorial interpretation for them. When $m=0$ these numbers are the middle binomial coefficients $\binom{2n}{n}$, and when $m=1$ they are twice the Catalan numbers. In this paper, we give combinatorial interpretations for these numbers wh...

The original derivation of a 19th century identity associated with K. von Szily is presented and discussed. An independent proof is given using a technique developed a decade or so later (by J. Dougall) in relation to hypergeometric function theory. For completeness, a historical backdrop is provided for the reader, together with other relevant inf...

We describe applications of the classical umbral calculus to bilinear generating functions for polynomial sequences, identities for Bernoulli and related numbers, and Kummer congruences.

We count labeled acyclic digraphs according to the number sources, sinks, and edges. 1. Counting acyclic digraphs by sources. Let A n (t; #)= s(D) where the sum is over all acyclic digraphs D on the vertex set [n]={1, 2,... ,n}, e(D)isthe number of edges of D, and s(D)isthe number of sources of D; that is, the number of vertices of D of indegree 0....

A simple decomposition for graphs yields generating functions for counting graphs by edges and connected components. A change of variables gives a new interpretation to the Tutte polynomial of the complete graph involving inversions of trees. The relation between the Tutte polynomial of the complete graph and the inversion enumerator for trees is g...

We derive new functional equations at a species level for certain classes of 2-trees, including a dissymmetry theorem. From these equations we deduce various series expansions for these structures. We obtain formulas for unlabeled 2-trees which are more explicit than previously known results. Moreover, the asymptotic behavior of unlabeled 2-trees i...

We prove that the order of divisibility by prime p of k! S(a (p − 1) pq ,k )d oesnot depend on a and q if q is sufficiently large and k/p is not an odd integer. Here S(n, k) denotes the Stirling number of the second kind; i.e., the number of partitions of a set of n objects into k nonempty subsets. The proof is based on divisibility results for p-s...

Given a set of positive integers A = {a 1 ,...,a n }, we study the number p A (t)of nonnegative integer solutions (m 1 ,...,m n )to # n j=1 m j a j = t. We derive an explicit formula for the polynomial part of p A . Let A = {a 1 ,...,a n } be a set of positive integers with gcd(a 1 ,...,a n )=1. The classical Frobenius problem asks for the largest...

During the 1880s Eugène Catalan disseminated an interesting identity involving Catalan numbers (at the time called Segner numbers by him) which today seems to have become forgotten. Further to a recent account of his derivation of it, we here present modern proofs created through the mathematics of generalised hypergeometric series.

this article is to explain how generating functions arise naturally in enumeration problems.

Introduction In studying representability of matroids, Lindstrom [42] gave a combinatorial interpretation to certain determinants in terms of disjoint paths in digraphs. In a previous paper [25], the authors applied this theorem to determinants of binomial coe#cients. Here we develop further applications. As in [25], the paths under consideration a...

This paper consists of two related parts. In the first part the theory of D-finite power series in several variables and the theory of symmetric functions are used to prove P-recursiveness for regular graphs and digraphs and related objects, that is, that their counting sequences satisfy linear homogeneous recurrences with polynomial coefficients....

We derive new functional equations at a species level for 2trees including a new dissymmetry theorem. From these equations we deduce various enumerative series expansions for certain classes of 2-trees, including the molecular decomposition in some cases.

The r-th Faber polynomial of the Laurent series f(t) = t + f 0 + f 1/t + f 2/t 2 + … is the unique polynomial F r (u) of degree r in u such that F r (f) = t r + negative powers of t. We apply Faber polynomials, which were originally used to study univalent functions, to lattice path enumeration.

. Lin and Chang gave a generating function for the number of convex polyominoes with an m+1byn+ 1 minimal bounding rectangle. We show that their result implies that the number of such polyominoes is m+ n +mn m+ n # 2m +2n 2m # - 2(m + n) # m+ n - 1 m ## m+ n - 1 n # . Resum e. Lin et Chang ont donnelaserie generatrice du nombre de polyominos convex...

this paper is to analyze # p (k! S(n, k)) for an arbitrary prime p. It turns out that identity (1) can be generalized to calculate the exact value of # p (k! S(n, k)) if n =

rticular we shall relate the number of passes directly to one of these: A n,k is the number of permutations # of {1,...,n} with k strong excedances, which are values of i for which i<#(i). Before we take up the Smith College diploma problem, we explain the connection between this interpretation of the Eulerian numbers and one that is somewhat This...

. Let P (k) be the chromatic polynomial of a graph with n # 2 vertices and no isolated vertices, and let R(k)=P (k +1)/k(k + 1). We show that the coe#cients of the polynomial (1 - t) n-1 # # k=1 R(k)t k are nonnegative and we give a combinatorial interpretation to R(k) when k is a nonpositive integer. 1. Introduction. The problem of characterizing...

We count the pairs of walks between diagonally opposite corners of a given lattice rectangle by the number of points in which they intersect. We note that the number of such pairs with one intersection is twice the number with no intersection, and we give a bijective proof of that fact. Some probabilistic variants of the problem are also investigat...

We show how to count tilings of Aztec diamonds and hexagons with defects using determinants. In several cases these determinants can be evaluated in closed form. In particular, we obtain solutions to open problems 1, 2, and 10 in James Propp's list of problems on enumeration of matchings [22]. 1. Introduction While studying dimer models, P. W. Kast...

One of the most important numerical quantities that can be computed from a graph G is the two-variable Tutte polynomial. Specializations of the Tutte polynomial count various objects associated with G, e.g., subgraphs, spanning trees, acyclic orientations, inversions and parking functions. We show that by partitioning certain simplicial complexes r...

We study sums of the form R(#), where R is a rational function and the sum is over all nth roots of unity # (often with # = 1 excluded). We call these generalized Dedekind sums, since the most well-known sums of this form are Dedekind sums. We discuss three methods for evaluating such sums: The method of factorization applies if we have an explicit...

Abstract We identify a set of d! signed points, called Toeplitz points , in Zd , with the following property: for every n > 0, the excess of the number of lattice walks of n steps, from the origin to all positive Toeplitz points, over the number to all negative Toeplitz points, is equal to n times the number of permutations of f1; 2; : : : ; ng tha...

A descent of a rooted tree with totally ordered vertices is a vertex that is greater than at least one of its children. A leaf is a vertex with no children. We show that the number of forests of rooted trees on a given vertex set with i +1 leaves and j descents is equal to the number with j +1 leaves and i descents. We do this by finding a function...

We count labeled acyclic digraphs according to the number sources, sinks, and edges.

Let M(r,s)(n,k) be the number of ordered pairs of paths in the plane, with unit steps E or N, that intersect k times in which the first path ends al the point (r,n-r) and the second path ends at the point (s,n-s). Let N-E(n, k, p) = (r+s=2p)Sigma M(r,s)(n,k) and N-O(n, k, p) = (r+s=2p+1)Sigma M(r,s)(n,k). We study the numbers M(r,s)(n,k), N-k(n,r)...

A priority queue transforms an input permutation p 1 of a totally ordered set of size n into an output permutation p 2 . Atkinson and Thiyagarajah showed that the number of such pairs (p 1 ; p 2 ) is (n+1) nGamma1 , which is well known to be the number of labeled trees on n + 1 vertices. We give a new proof of this result by finding a bijection fro...

. A new object is introduced into the theory of partitions that generalizes plane partitions: cylindric partitions. We obtain the generating function for cylindric partitions of a given shape that satisfy certain row bounds as a sum of determinants of q-binomial coefficients. In some special cases these determinants can be evaluated. Extending an i...

Extending the work of Wilf and Zeilberger on WZ-pairs, we describe how new terminating hypergeometric series identities can be derived by duality from known identities. A large number of such identities are obtained by a Maple program that applies this method systematically.

We give a simple combinatorial proof a Langrange inversion theorem for species and derive from it Labelle's Lagrange inversion theorem for cycle index series (or equivalently, symmetric functions). We also discuss the combinatorial meaning of the differential operators that arise in various forms of Lagrange inversion.

Mallows and Riordan “The Inversion Enumerator for Labeled Trees,” Bulletin of the American Mathematics Society, vol. 74 [1968] pp. 92-94) first defined the inversion polynomial, Jn(q) for trees with n vertices and found its generating function. In the present work, we define inversion polynomials for ordered, plane, and cyclic trees, and find their...

One of the most important numerical quantities that can be computed from a graph $G$ is the two-variable Tutte polynomial. Specializations of the Tutte polynomial count various objects associated with $G$, e.g., subgraphs, spanning trees, acyclic orientations, inversions and parking functions. We show that by partitioning certain simplicial complex...