Distributions of mesh patterns of short lengths
Abstract
A systematic study of avoidance of mesh patterns of length 2 was conducted by Hilmarsson et al., where 25 out of 65 non-equivalent cases were solved. In this paper, we give 27 distribution results for these patterns including 14 distributions for which avoidance was not known. Moreover, for the unsolved cases, we prove an equidistribution result (out of 6 equidistribution results we prove in total), and conjecture 6 more equidistributions. Finally, we find seemingly unknown distribution of the well known permutation statistic ``strict fixed point'', which plays a key role in many of our enumerative results. This paper is the first systematic study of distributions of mesh patterns. Our techniques to obtain the results include, but are not limited to, obtaining functional relations for generating functions, and finding recurrence relations and bijections.
Keywords
Cite
@article{arxiv.1811.07679,
title = {Distributions of mesh patterns of short lengths},
author = {Sergey Kitaev and Philip B. Zhang},
journal= {arXiv preprint arXiv:1811.07679},
year = {2019}
}
Comments
Advances in Applied Mathematics 110 (2019) 1-32; there is a mistake in the proof of Theorem 5.1 in version 1 of the paper; its statement is now a conjecture