On Lattice Path Counting by Major Index and Descents

Abstract

A formula for counting lattice paths in the plane from μ = (μ1, μ2) to λ = (λ1, λ2) which do not cross the lines y = x + d and y = x + c, where c, d ϵ Z and d >c, by descents and major index is given. The proof, which is purely combinatorial, uses a bijection on certain two-rowed tableaux. As applications, formulas for the joint distribution of Kolmogorov-Smirnov and run statistics are derived.