Joint distributions of statistics over permutations avoiding two patterns of length 3

Han, Tian and Kitaev, Sergey (2024) Joint distributions of statistics over permutations avoiding two patterns of length 3. Discrete Mathematics and Theoretical Computer Science. ISSN 1365-8050 (In Press)

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Abstract

Finding distributions of permutation statistics over pattern-avoiding classes of permutations attracted much attention in the literature. In particular, Bukata et al.~¥cite{Bukata2019} found distributions of ascents and descents on permutations avoiding any two patterns of length 3. In this paper, we generalize these results in two different ways: we find explicit formulas for the joint distribution of six statistics ($¥asc$, $¥des$, $¥lrmax$, $¥lrmin$, $¥rlmax$, $¥rlmin$), and also explicit formulas for the joint distribution of four statistics ($¥asc$, $¥des$, $¥MNA$, $¥MND$) on these permutations in all cases. The latter result also extends the recent studies by Kitaev and Zhang~¥cite{KitZha} of the statistics $¥MNA$ and $¥MND$ (related to non-overlapping occurrences of ascents and descents) on stack-sortable permutations. All multivariate generating functions in our paper are rational, and we provide combinatorial proofs of five equidistribution results that can be derived from the generating functions.

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