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Ira M. Gessel

Publications and source records attributed to Ira M. Gessel.

At least 37 records · Page 2Linked to original sources
arXiv2018

On the Schur function expansion of a symmetric quasi-symmetric function

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 new proof. We give here a simple proof of Garsia and Remmel's version, using a sign-reversing involution.

math.CO↗
arXiv2016

Lagrange Inversion

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

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arXiv2015

A historical survey of P-partitions

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.

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arXiv2014

Counting permutations by alternating descents

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 uses a system of differential equations. The second proof derives the generating function directly from general permutation enumeration techniques, using noncommutative symmetric functions. The generating function is an "alternating" analogue of David and Barton's generating function for permutations with no increasing runs of length 3 or more. Our general results give further alternating analogues of permutation enumeration formulas, including results of Chebikin and Remmel.

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arXiv2014

Counting unlabeled k-trees

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.

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arXiv2013

Non-orientable surfaces in homology cobordisms

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 Ozsváth--Sazbó $d$-invariants or Atiyah--Singer $ρ$-invariants of $M$. One consequence is that the minimal genus of a smoothly embedded surface in $L(2p,q) \times I$ is the same as the minimal genus of a surface in $L(2p,q)$. We also consider embeddings of non-orientable surfaces in closed $4$-manifolds.

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arXiv2013

Compositions and Fibonacci Identities

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

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arXiv2013

A simple proof of Andrews's 5F4 evaluation

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.

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arXiv2013

Even and Odd Pairs of Lattice Paths with Multiple Intersections

Let M(n,k,r,s) be the number of ordered paths in the plane, with unit steps E or N, that intersect k times in which the first path ends at the point (r,n-r) and the second path ends at the point (s,n-s). Our main object of study in this paper is the sum of the numbers M(n,k,r,s) over r and s where r+s is fixed. We consider even and odd values of r+s separately, and we derive a simpler formula for M(n,k,r,s) than previously appeared in the literature.

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arXiv2012

Jacobi-Stirling polynomials and $P$-partitions

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 $\JS(n+k,n;z)=p_{k,0}(n)+p_{k,1}(n)z+...+p_{k,k}(n)z^k$, we show that $(1-t)^{3k-i+1}\sum_{n\geq0}p_{k,i}(n)t^n$ is a polynomial in $t$ with nonnegative integral coefficients and provide combinatorial interpretations of the coefficients by using Stanley's theory of $P$-partitions.

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arXiv2011

Symmetrically Constrained Compositions

Given integers $a_1, a_2, ..., a_n$, with $a_1 + a_2 + ... + a_n \geq 1$, a symmetrically constrained composition $λ_1 + lambda_2 + ... + lambda_n = M$ of $M$ into $n$ nonnegative parts is one that satisfies each of the the $n!$ constraints ${\sum_{i=1}^n a_i λ_{π(i)} \geq 0 : π\in S_n}$. We show how to compute the generating function of these compositions, combining methods from partition theory, permutation statistics, and lattice-point enumeration.

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arXiv2010

A note on 2-distant noncrossing partitions and weighted Motzkin paths

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 combinatorial.

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arXiv2010

The limit as p -> infinity of the Hilbert-Kunz multiplicity of sum(x_i^(d_i))

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.

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arXiv2009

Multilinear generating functions for Charlier polynomials

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.

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arXiv2008

A Unified Elementary Approach to the Dyson, Morris, Aomoto, and Forrester Constant Term Identities

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 some special cases of the Forrester conjecture.

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