Ira M. Gessel

Binomial determinants, paths, and hook length formulae

Abstract We give a combinatorial interpretation for any minor (or binomial determinant) of the matrix of binomial coefficients. This interpretation involves configurations of nonintersecting paths, and is related to Young tableaux and hook length formulae.

Counting permutations with given cycle structure and descent set

Abstract The number of permutations with given cycle structure and descent set is shown to be equal to the scalar product of two special characters of the symmetric group. Enumerative applications are given to cycles, involutions and derangements.

A noncommutative generalization and

The Lagrange inversion formula is generalized to formal power series in noncommutative variables. A g-analog is obtained by applying a linear operator to the noncommutative formula before substituting commuting variables.

q

The classical Ballot problem that counts the number of ways of walking from the origin and staying within the wedge x 1 ≥x 2 ≥...≥x n (which is a Weyl chamber for the symmetric group), using positive unit steps, is generalized to general Weyl groups and general sets of steps

-analog of the Lagrange inversion formula

Many evaluations of terminating hypergeometric series at arguments other than 1 are given. Some are equivalent to some unpublished work of Gosper, while others are new. In particular, two new evaluations of

Random walk in a Weyl chamber

{}_7 F_6

Strange Evaluations of Hypergeometric Series

’s with four parameters are stated. The main technique is a change of variables formula which is equivalent to the Lagrange inversion formula. A new proof of Whipple’s transformation of a very well poised

A combinatorial proof of the multivariable Lagrange inversion formula

{}_7 F_6

A probabilistic method for lattice path enumeration

into a Saalschutzian

Some congruences for Apéry numbers

{}_4 F_3