Burstein's permutation conjecture, Hong and Li's inversion sequence conjecture, and restricted Eulerian distributions

TL;DR

This paper confirms conjectures relating pattern-avoiding inversion sequences and permutations, providing new generating function identities and establishing equidistribution results for certain statistics.

Contribution

It proves Hong and Li's conjecture on 0021-avoiding inversion sequences and Burstein's related permutation conjecture, along with new generating function identities and equidistribution results.

Findings

Confirmed Hong and Li's conjecture on 0021-avoiding inversion sequences.

Validated Burstein's permutation counting conjecture.

Derived new generating function identities and equidistribution results.

Abstract

Recently, Hong and Li launched a systematic study of length-four pattern avoidance in inversion sequences, and in particular, they conjectured that the number of $0021$-avoiding inversion sequences can be enumerated by the OEIS entry A218225. Meanwhile, Burstein suggested that the same sequence might also count three sets of pattern restricted permutations. The objective of this paper is not only a confirmation of Hong and Li's conjecture and Burstein's first conjecture, but also two more delicate generating function identities with the $\mathsf{ides}$ statistic concerned in the restricted permutation case, and the $\mathsf{asc}$ statistic concerned in the restricted inversion sequence case, which yield a new equidistribution result.