Signed Enumeration of Upper-Right Corners in Path Shuffles
William Kuszmaul

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 -positive. Finally, we conjecture an equivalence between -positivity of the generating function for upper-right-corner count and signed upper-right-corner count.
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