Signed Enumeration of Upper-Right Corners in Path Shuffles
Combinatorics
2016-10-31 v2
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.
Keywords
Cite
@article{arxiv.1510.00777,
title = {Signed Enumeration of Upper-Right Corners in Path Shuffles},
author = {William Kuszmaul},
journal= {arXiv preprint arXiv:1510.00777},
year = {2016}
}