Counting lattice paths by crossings and major index I: the corner-flipping bijections

TL;DR

This paper introduces bijections to enumerate lattice paths with steps (1,1) and (1,-1) based on crossings and major index, providing simple formulas involving q-binomial coefficients.

Contribution

It presents new bijections and formulas for counting lattice paths by crossings and major index, advancing combinatorial enumeration methods.

Findings

Derived simple formulas using q-binomial coefficients

Established bijections with visual descriptions

Solved enumeration problems related to crossings and major index

Abstract

We solve two problems regarding the enumeration of lattice paths in $\mathbb{Z}^2$ with steps $(1,1)$ and $(1,-1)$ with respect to the major index, defined as the sum of the positions of the valleys, and to the number of certain crossings. The first problem considers crossings of a single path with a fixed horizontal line. The second one counts pairs of paths with respect to the number of times they cross each other. Our proofs introduce lattice path bijections with convenient visual descriptions, and the answers are given by remarkably simple formulas involving $q$-binomial coefficients.