Skew Schur Function Representation of Directed Paths in a Slit

Abstract

In this work, we establish a general relationship between the enumeration of weighted directed paths and skew Schur functions, extending work by Bousquet-Mélou, who expressed generating functions of discrete excursions in terms of rectangular Schur functions.

Length and area generating functions for height-restricted Motzkin meanders

“…The expressions of length generating functions (q = 1) for Dyck, Motzkin and more general paths with a number of possible up and down steps, and arbitrary weights associated to each kind of step, have been related to skew-Schur functions [17]. Including the area counting variable q would generalize these generating functions to q-deformed versions of skew-Schur functions, both in the case of Dyck paths and for Motzkin paths.…”

“…In recent work [10] the Hamiltonian description of random walks and the exclusion statistics connection were used to study the generating function of a family of walks referred to as Dyck paths and their height-restricted generalizations [11]− [17] 1 . These are walks on a two-dimensional lattice that propagate one step in the horizontal direction ("time") and one step either up or down in the vertical direction ("height") but without dipping below a "floor" at height zero nor exceeding a "ceiling" of maximal height.…”