Signed Enumeration of Upper-Right Corners in Path Shuffles

TL;DR

This paper resolves a conjecture on enumerating quarter-plane walks by their upper-right-corner count modulo 2, introduces a signed count statistic, and explores its distribution and positivity properties.

Contribution

It introduces a signed upper-right-corner count statistic and proves its distribution and positivity properties in planar walks with fixed projections.

Findings

Distribution of signed upper-right-corner count over planar walks

Polynomial counting loops is (x+1)-positive

Conjecture on equivalence between positivity and signed count

Abstract

We resolve a conjecture of Albert and Bousquet-Melou enumerating quarter-plane walks with fixed horizontal and vertical projections according to their upper-right-corner count modulo 2. In doing this, we introduce a signed upper-right-corner count statistic. We find its distribution over planar walks with any choice of fixed horizontal and vertical projections. Additionally, we prove that the polynomial counting loops with a fixed horizontal and vertical projection according to the absolute value of their signed upper-right-corner count is $(x+1)$-positive. Finally, we conjecture an equivalence between $(x+1)$-positivity of the generating function for upper-right-corner count and signed upper-right-corner count.