## 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 language | English |
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Article number | 4 |

Journal | Discrete Mathematics and Theoretical Computer Science |

Volume | 24 |

Issue number | 1 |

DOIs | |

State | Published - 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