arXiv subjects

Ira M. Gessel

Publications and source records attributed to Ira M. Gessel.

At least 19 recordsLinked to original sources

arXiv2025

Lattice paths and the Geode

Let $t_1,t_2,\dots$ be variables, and let $S$ be the formal power series in the variables $t_1, t_2,\dots$ satisfying $S=1+\sum_{i=1}^\infty t_n S^n.$ Let $S_1 =\sum_{n=1}^\infty t_n$. Wildberger and Rubine recently showed that there is a formal power series $G$ in the $t_i$, which they called the Geode, satisfying $S=1+GS_1$. In this paper we discuss some of the properties of the Geode and of the related series $H=G/S$, which satisfies $S=1/(1-HS_1)$. We show that \begin{equation*} G=\biggl(1-\sum_{n=1}^\infty t_n (1+S+S^2+\cdots+S^{n-1})\biggr)^{-1}, \end{equation*} and \begin{equation*} H=\biggl( 1-\sum_{n=2}^\infty t_n (S+S^2+\cdots+S^{n-1})\biggr)^{-1}, \end{equation*} and we give combinatorial interpretations of $G$ and $H$ in terms of lattice paths.

math.CO↗

arXiv2024

Counting up-up-or-down-down permutations

Answering a question of Donald Knuth, we find the bivariate exponential generating function for "up-up-or-down-down'' permutations of odd length according to their last entry. An up-up-or-down-down permutation is a permutation $a_1a_2\cdots a_n$ satisfying $a_{2i-1}<a_{2i}$ if and only if $a_{2i}<a_{2i+1}$ for $1\le i <n/2$. Equivalently, an up-up-or-down-down permutation is one in which every peak and every valley is odd.

math.CO↗

arXiv2024

Two-sided permutation statistics via symmetric functions

Given a permutation statistic $\operatorname{st}$, define its inverse statistic $\operatorname{ist}$ by $\operatorname{ist}(π):=\operatorname{st}(π^{-1})$. We give a general approach, based on the theory of symmetric functions, for finding the joint distribution of $\operatorname{st}_{1}$ and $\operatorname{st}_{2}$ whenever $\operatorname{st}_{1}$ and $\operatorname{st}_{2}$ are descent statistics: permutation statistics that depend only on the descent composition. We apply this method to a number of descent statistics, including the descent number, the peak number, the left peak number, the number of up-down runs, and the major index. Perhaps surprisingly, in many cases the polynomial giving the joint distribution of $\operatorname{st}_{1}$ and $\operatorname{ist}_{2}$ can be expressed as a simple sum involving products of the polynomials giving the (individual) distributions of $\operatorname{st}_{1}$ and $\operatorname{st}_{2}$. Our work leads to a rederivation of Stanley's generating function for doubly alternating permutations, as well as several conjectures concerning real-rootedness and $γ$-positivity.

math.CO↗

arXiv2024

Binomial convolutions for rational power series

The binomial convolution of two sequences $\{a_n\}$ and $\{b_n\}$ is the sequence whose $n$th term is $\sum_{k=0}^{n} \binom{n}{k} a_k b_{n-k}$. If $\{a_n\}$ and $\{b_n\}$ have rational generating functions then so does their binomial convolution. We discuss an efficient method, using resultants, for computing this rational generating function and give several examples involving Fibonacci and tribonacci numbers and related sequences. We then describe a similar method for computing Hadamard products of rational generating functions. Finally we describe two additional methods for computing binomial convolutions and Hadamard products of rational power series, one using symmetric functions and one using partial fractions.

math.CO↗

arXiv2023

A short proof of the Almkvist-Meurman theorem

We give a short generating function proof of the Almkvist-Meurman theorem: For integers $h$ and $k\ne0$, define the numbers $M_n(h,k)$ by $kx(e^{hx}-1)/(e^{kx}-1)=\sum_{n=0}^\infty M_n(h,k) x^n/n!$. Equivalently, $M_n(h,k) = k^n(B_n(h/k) - B_n)$, where $B_n(u)$ is the Bernoulli polynomial. Then $M_n(h,k)$ is an integer. The proof is related to Postnikov's functional equation for the generating function for intransitive trees.

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arXiv2023

Good Will Hunting's Problem: Counting Homeomorphically Irreducible Trees

In the film Good Will Hunting, the main character, a janitor at MIT named Will Hunting, attacks the problem of drawing all the homeomorphically irreducible trees with 10 vertices. Although the film suggests that this is a difficult problem, it is in fact quite easy. A much more interesting problem is counting homeomorphically irreducible trees with $n$ vertices for all $n$, a feat accomplished by Harary and Prins in 1959. Here we give an exposition and simplification of Harary and Prins's result, introducing some of the fundamental ideas of graphical enumeration.

math.CO↗

arXiv2023

On the Almkvist-Meurman theorem for Bernoulli polynomials

Almkvist and Meurman showed that if h and k are integers, then so is $k^n(B_n(h/k) - B_n)$ where $B_n(u)$ is the Bernoulli polynomial. We give here a new and simpler proof of the Almkvist-Meurman theorem using generating functions. We describe some properties of these numbers and prove a common generalization of the Almkvist-Meurman theorem and a result of Gy on Bernoulli-Stirling numbers. We then give a simple generating function proof of an analogue of the Almkvist-Meurman theorem for Euler polynomials, due to Fox.

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arXiv2022

An application of the Goulden-Jackson cluster theorem

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 lattices with Möbius function 0, 1, or -1.

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arXiv2022

Moments of Orthogonal Polynomials and Exponential Generating Functions

Starting from the moment sequences of classical orthogonal polynomials we derive the orthogonality purely algebraically. We consider also the moments of ($q=1$) classical orthogonal polynomials, and study those cases in which the exponential generating function has a nice form. In the opposite direction, we show that the generalized Dumont-Foata polynomials with six parameters are the moments of rescaled continuous dual Hahn polynomials. Finally we show that one of our methods can be applied to deal with the moments of Askey-Wilson polynomials.

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arXiv2021

Counting tanglegrams with species

A tanglegram is a pair of binary trees with the same set of leaves. Unlabeled tanglegrams were counted recently by Billey, Konvalinka, and Matsen, who also proposed the problem of counting several variations of unlabeled tanglegrams (unordered and unrooted tanglegrams). We use the theory of combinatorial species to solve these problems.

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arXiv2021

On a polynomial congruence for Eulerian polynomials

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.

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arXiv2020

Plethystic formulas for permutation enumeration

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 number, peak number, left peak number, and the number of up-down runs. We apply these results to cyclic permutations, involutions, and derangements, and more generally, to derive formulas for counting all permutations by the above statistics jointly with the number of fixed points and jointly with cycle type. A number of known formulas are recovered as special cases of our results, including formulas of Désarménien-Foata, Gessel-Reutenauer, Stembridge, Fulman, Petersen, Diaconis-Fulman-Holmes, Zhuang, and Athanasiadis.

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arXiv2020

Cyclic quasi-symmetric functions

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 cyclic shuffles of permutations. We then prove the following positivity phenomenon: for every non-hook shape $λ$, the coefficients in the expansion of the Schur function $s_λ$ in terms of fundamental cyclic quasi-symmetric functions are nonnegative. The proof relies on the existence of a cyclic descent map on the standard Young tableaux (SYT) of shape $λ$. The theory has applications to the enumeration of cyclic shuffles and SYT by cyclic descents.

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arXiv2020

A note on Stirling permutations

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

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arXiv2020

Counting acyclic and strong digraphs by descents

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 function.

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arXiv2019

Labeled binary trees, subarrangements of the Catalan arrangements, and Schur positivity

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 using a weight-preserving bijection whose inverse is inspired by the Push-Glide algorithm of Préville-Ratelle and Viennot. In fact, this weight-preserving bijection allows us to establish a stronger version of the first author's conjecture showing that the generating function restricted to labeled binary trees with a fixed canopy is still Schur positive. We also discuss applications in the setting of hyperplane arrangements. We show that a certain specialization of $G$ equals the Frobenius characteristic of the natural $\mathfrak{S}_n$-action on regions of the semiorder arrangement, which we then expand in terms of {the Frobenius characteristics} of Foulkes characters. We also construct an $\mathfrak{S}_n$-action on regions of the Linial arrangement using a set of trees studied by Bernardi, and subsequently compute the character of this action by employing Lagrange inversion. The resulting expression generalizes Postnikov's formula for the number of regions in the Linial arrangement. As a final application, we prove $γ$-nonnegativity for the distribution of the number of right edges over local binary search trees.

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arXiv2019

Reciprocals of exponential polynomials and permutation enumeration

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 are restricted to certain congruence classes, and extend these results to noncommutative symmetric functions that count words with the same restrictions on run lengths.

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arXiv2018

Shuffle-compatible permutation statistics

Since the early work of Richard Stanley, it has been observed that several permutation statistics have a remarkable property with respect to shuffles of permutations. We formalize this notion of a shuffle-compatible permutation statistic and introduce the shuffle algebra of a shuffle-compatible permutation statistic, which encodes the distribution of the statistic over shuffles of permutations. This paper develops a theory of shuffle-compatibility for descent statistics (statistics that depend only on the descent set and length) which has close connections to the theory of $P$-partitions, quasisymmetric functions, and noncommutative symmetric functions. We use our framework to prove that many descent statistics are shuffle-compatible and to give explicit descriptions of their shuffle algebras, thus unifying past results of Stanley, Gessel, Stembridge, Aguiar-Bergeron-Nyman, and Petersen.

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