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arXiv · 1510.00777

Signed Enumeration of Upper-Right Corners in Path Shuffles

William Kuszmaul
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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.

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William Kuszmaul. 2016-10-28. Signed Enumeration of Upper-Right Corners in Path Shuffles. https://arxiv.org/abs/1510.00777

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