Ira Gessel - Academia.edu (original) (raw)

Papers by Ira Gessel

Research paper thumbnail of Enumeration of tilings of diamonds and hexagons with defects

Electr J Comb, 1998

We show how to count tilings of Aztec diamonds and hexagons with defects using determinants. In s... more 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 problems 1, 2, and 10 in James Propp's list of problems on enumeration of matchings.

Research paper thumbnail of Multipartite P-partitions and inner products of skew Schur functions

Research paper thumbnail of Communicated by the Managing Editors

Research paper thumbnail of Tournaments and Vandermonde''''s determinant

Research paper thumbnail of On the Descent Numbers and Major Indices for the Hyperoctahedral Group

, in answering a question posed by Foata, introduced two descent numbers and major indices for th... more , in answering a question posed by Foata, introduced two descent numbers and major indices for the hyperoctahedral group B n , whose joint distribution generalizes an identity due to MacMahon and Carlitz. We shall show that yet another pair of statistics exists, and whose joint distribution constitutes a "natural" solution to Foata's problem.

Research paper thumbnail of Counting three-line Latin rectangles

Research paper thumbnail of Divisibility properties of the -tangent numbers

Proceedings of the American Mathematical Society, 1978

Research paper thumbnail of Multipartite 𝑃-partitions and inner products of skew Schur functions

Contemporary Mathematics, 1984

Research paper thumbnail of Book Review: Enumerative combinatorics, Volume 2

Research paper thumbnail of Lattice Paths and Faber Polynomials

Advances in Combinatorial Methods and Applications to Probability and Statistics, 1997

The rth Faber polynomial of the Laurent series f (t) = t + f 0 + f 1 /t + f 2 /t 2 + · · · is the... more The rth 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.

Research paper thumbnail of Book Review: Combinatorial enumeration

Bulletin of the American Mathematical Society, 1985

Research paper thumbnail of Generalized Rook Polynomials and Orthogonal Polynomials

The IMA Volumes in Mathematics and Its Applications, 1989

We consider several generalizations of rook polynomials. In particular we develop analogs of the ... more We consider several generalizations of rook polynomials. In particular we develop analogs of the theory of rook polynomials that are related to general Laguerre and Charlier polynomials in the same way that ordinary rook polynomials are related to simple Laguerre polynomials.

Research paper thumbnail of The Smith College Diploma Problem

Research paper thumbnail of Applications of the classical umbral calculus

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

Research paper thumbnail of The Tutte Polynomial of a Graph, Depth-first Search

The Electronic Journal of Combinatorics, Jun 14, 1995

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

Eprint Arxiv 1007 2004, Jul 12, 2010

Let p be a prime. The Hilbert-Kunz multiplicity, mu, of the element sum(x_i^(d_i)) of (Z/p)[x_1,.... more 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.

Research paper thumbnail of Counting permutations by alternating descents

The Electronic Journal of Combinatorics, Aug 8, 2014

We find the exponential generating function for permutations with all valleys even and all peaks ... more 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.

Research paper thumbnail of An Introduction to Lattice Path Enumeration

Page 1. AN INTRODUCTION TO LATTICE PATH ENUMERATION IRA M. GESSEL Abstract. ... [3] A. Dvoretzky ... more Page 1. AN INTRODUCTION TO LATTICE PATH ENUMERATION IRA M. GESSEL Abstract. ... [3] A. Dvoretzky and Th. Motzkin, A problem of arrangements, Duke Math. J. 14 (1947), 305–313. [4] J. Peacock, On “Al Capone and the Death Ray”, Math. Gazette 26 (1942), 218–219. ...

Research paper thumbnail of On the Order of Stirling Numbers and Alternating Binomial Coefficient Sums

Fibonacci Quarterly, 2001

We prove that the p-adic order of k! S(a (p − 1) p q ,k) does not depend on a and q if p − 1 | k ... more We prove that the p-adic order of k! S(a (p − 1) p q ,k) does not depend on a and q if p − 1 | k and q is sufficiently large. Here S(n, k) denotes the Stirling number of the second kind. The proof is based on divisibility results for p-sected alternating binomial coefficient sums. A fairly general criterion is also given to obtain divisibility properties of recurrent sequences when the coefficients follow some divisibility patterns.

Research paper thumbnail of On the Number of Convex Polyominoes

Lin and Chang gave a generating function for the number of convex polyominoes with an m + 1 by n ... more Lin and Chang gave a generating function for the number of convex polyominoes with an m + 1 by n + 1 minimal bounding rectangle. We show that their result implies that the number of such polyominoes is

Research paper thumbnail of Enumeration of tilings of diamonds and hexagons with defects

Electr J Comb, 1998

We show how to count tilings of Aztec diamonds and hexagons with defects using determinants. In s... more 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 problems 1, 2, and 10 in James Propp's list of problems on enumeration of matchings.

Research paper thumbnail of Multipartite P-partitions and inner products of skew Schur functions

Research paper thumbnail of Communicated by the Managing Editors

Research paper thumbnail of Tournaments and Vandermonde''''s determinant

Research paper thumbnail of On the Descent Numbers and Major Indices for the Hyperoctahedral Group

, in answering a question posed by Foata, introduced two descent numbers and major indices for th... more , in answering a question posed by Foata, introduced two descent numbers and major indices for the hyperoctahedral group B n , whose joint distribution generalizes an identity due to MacMahon and Carlitz. We shall show that yet another pair of statistics exists, and whose joint distribution constitutes a "natural" solution to Foata's problem.

Research paper thumbnail of Counting three-line Latin rectangles

Research paper thumbnail of Divisibility properties of the -tangent numbers

Proceedings of the American Mathematical Society, 1978

Research paper thumbnail of Multipartite 𝑃-partitions and inner products of skew Schur functions

Contemporary Mathematics, 1984

Research paper thumbnail of Book Review: Enumerative combinatorics, Volume 2

Research paper thumbnail of Lattice Paths and Faber Polynomials

Advances in Combinatorial Methods and Applications to Probability and Statistics, 1997

The rth Faber polynomial of the Laurent series f (t) = t + f 0 + f 1 /t + f 2 /t 2 + · · · is the... more The rth 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.

Research paper thumbnail of Book Review: Combinatorial enumeration

Bulletin of the American Mathematical Society, 1985

Research paper thumbnail of Generalized Rook Polynomials and Orthogonal Polynomials

The IMA Volumes in Mathematics and Its Applications, 1989

We consider several generalizations of rook polynomials. In particular we develop analogs of the ... more We consider several generalizations of rook polynomials. In particular we develop analogs of the theory of rook polynomials that are related to general Laguerre and Charlier polynomials in the same way that ordinary rook polynomials are related to simple Laguerre polynomials.

Research paper thumbnail of The Smith College Diploma Problem

Research paper thumbnail of Applications of the classical umbral calculus

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

Research paper thumbnail of The Tutte Polynomial of a Graph, Depth-first Search

The Electronic Journal of Combinatorics, Jun 14, 1995

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

Eprint Arxiv 1007 2004, Jul 12, 2010

Let p be a prime. The Hilbert-Kunz multiplicity, mu, of the element sum(x_i^(d_i)) of (Z/p)[x_1,.... more 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.

Research paper thumbnail of Counting permutations by alternating descents

The Electronic Journal of Combinatorics, Aug 8, 2014

We find the exponential generating function for permutations with all valleys even and all peaks ... more 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.

Research paper thumbnail of An Introduction to Lattice Path Enumeration

Page 1. AN INTRODUCTION TO LATTICE PATH ENUMERATION IRA M. GESSEL Abstract. ... [3] A. Dvoretzky ... more Page 1. AN INTRODUCTION TO LATTICE PATH ENUMERATION IRA M. GESSEL Abstract. ... [3] A. Dvoretzky and Th. Motzkin, A problem of arrangements, Duke Math. J. 14 (1947), 305–313. [4] J. Peacock, On “Al Capone and the Death Ray”, Math. Gazette 26 (1942), 218–219. ...

Research paper thumbnail of On the Order of Stirling Numbers and Alternating Binomial Coefficient Sums

Fibonacci Quarterly, 2001

We prove that the p-adic order of k! S(a (p − 1) p q ,k) does not depend on a and q if p − 1 | k ... more We prove that the p-adic order of k! S(a (p − 1) p q ,k) does not depend on a and q if p − 1 | k and q is sufficiently large. Here S(n, k) denotes the Stirling number of the second kind. The proof is based on divisibility results for p-sected alternating binomial coefficient sums. A fairly general criterion is also given to obtain divisibility properties of recurrent sequences when the coefficients follow some divisibility patterns.

Research paper thumbnail of On the Number of Convex Polyominoes

Lin and Chang gave a generating function for the number of convex polyominoes with an m + 1 by n ... more Lin and Chang gave a generating function for the number of convex polyominoes with an m + 1 by n + 1 minimal bounding rectangle. We show that their result implies that the number of such polyominoes is