Lattice walks in Z d and permutations with no long ascending subsequences

Abstract

We identify a set of d! signed points, called Toeplitz points, in Z d, 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/2n) times the number of permutation of {1,2,...,n} that contain no ascending subsequence of length > d. We prove this first by generating functions, using a determinantal theorem of Gessel. We give a second proof by direct construction of an appropriate involution. The latter provides a purely combinatorial proof of Gessel's theorem by interpreting it in terms of lattice walks. Finally we give a proof that uses the Schensted algorithm.