Further enumeration results concerning a recent equivalence of restricted inversion sequences

Let asc and desc denote respectively the statistics recording the number of ascents or descents in a sequence having non-negative integer entries. In a recent paper by Andrews and Chern, it was shown that the distribution of asc on the inversion sequence avoidance class $I_n(\geq,\neq,>)$ is the...

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Vydáno v:	Discrete Mathematics and Theoretical Computer Science Ročník 24, no. 1; číslo Combinatorics; s. 1 - 28
Hlavní autoři:	Mansour, Toufik, Shattuck, Mark
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Nancy DMTCS 01.01.2022 Discrete Mathematics & Theoretical Computer Science
Témata:	05a05 2021. 2010 mathematics subject classification. 05a15 65q20 65q30 pattern avoidance [math.math-co]mathematics [math]/combinatorics [math.co] [math.math-mp]mathematics [math]/mathematical physics [math-ph] Algorithms combinatorial statistic Combinatorics Enumeration Equivalence Functional equations inversion sequence Inversions (Geometry) july 22 kernel method Mathematical Physics Mathematical research Mathematics Sequences Sequences (Mathematics) Israel 2021. 2010 Mathematics Subject Classification. 05A15 65Q30 pattern avoidance 65Q20 July 22 kernel method 05A05 inversion sequence combinatorial statistic
ISSN:	1365-8050, 1462-7264, 1365-8050
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Popis
Shrnutí:	Let asc and desc denote respectively the statistics recording the number of ascents or descents in a sequence having non-negative integer entries. In a recent paper by Andrews and Chern, it was shown that the distribution of asc on the inversion sequence avoidance class $I_n(\geq,\neq,>)$ is the same as that of $n-1-\text{asc}$ on the class $I_n(>,\neq,\geq)$, which confirmed an earlier conjecture of Lin. In this paper, we consider some further enumerative aspects related to this equivalence and, as a consequence, provide an alternative proof of the conjecture. In particular, we find recurrence relations for the joint distribution on $I_n(\geq,\neq,>)$ of asc and desc along with two other parameters, and do the same for $n-1-\text{asc}$ and desc on $I_n(>,\neq,\geq)$. By employing a functional equation approach together with the kernel method, we are able to compute explicitly the generating function for both of the aforementioned joint distributions, which extends (and provides a new proof of) the recent result $\|I_n(\geq,\neq,>)\|=\|I_n(>,\neq,\geq)\|$. In both cases, an algorithm is formulated for computing the generating function of the asc distribution on members of each respective class having a fixed number of descents.
Bibliografie:	SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 14
ISSN:	1365-8050 1462-7264 1365-8050
DOI:	10.46298/dmtcs.8330