Further enumeration results concerning a recent equivalence of restricted inversion sequences

Toufik Mansour, Mark Shattuck

Research output: Contribution to journalArticlepeer-review

Abstract

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 In(≥, ≠, >) is the same as that of n − 1 − asc on the class In(>, ≠, ≥), 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 In(≥, ≠, >) of asc and desc along with two other parameters, and do the same for n − 1 − asc and desc on In(>, ≠, ≥). 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 that the common cardinality of In(≥, ≠, >) and In(>, ≠, ≥) is the same as that of Sn(4231, 42513). 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.

Original languageEnglish
Article number4
JournalDiscrete Mathematics and Theoretical Computer Science
Volume24
Issue number1
DOIs
StatePublished - 2022

Bibliographical note

Publisher Copyright:
© 2022 by the author(s)

Keywords

  • Combinatorial statistic
  • Inversion sequence
  • Kernel method
  • Pattern avoidance

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science (all)
  • Discrete Mathematics and Combinatorics

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