Combinatorics of the symmetries of ascents in restricted inversion sequences

TL;DR

This paper proves bijective symmetry and unimodality results for ascent distributions in certain restricted inversion sequences, advancing understanding of their combinatorial structure.

Contribution

It establishes bijective proofs for Lin's symmetry conjecture and the gamma-positivity of ascent polynomials in specific classes of inversion sequences.

Findings

Proves bijective symmetry of ascent distributions in restricted inversion sequences.

Shows gamma-positivity and unimodality of the ascent polynomial.

Provides combinatorial insights into the structure of inversion sequences avoiding certain triples.

Abstract

The systematic study of inversion sequences avoiding triples of relations was initiated by Martinez and Savage. For a triple $(\rho_1,\rho_2,\rho_3)\in\{<,>,\leq,\geq,=,\neq,-\}^3$, they introduced $\I_n(\rho_1,\rho_2,\rho_3)$ as the set of inversion sequences $e=e_1e_2\cdots e_n$ of length $n$ such that there are no indices $1\leq i<j<k\leq n$ with $e_i \rho_1 e_j$, $e_j \rho_2 e_k$ and $e_i \rho_3 e_k$. To solve a conjecture of Martinez and Savage, Lin constructed a bijection between $\I_n(\geq,\neq,>)$ and $\I_n(>,\neq,\geq)$ that preserves the distinct entries and further posed a symmetry conjecture of ascents on these two classes of restricted inversion sequences. Concerning Lin's symmetry conjecture, an algebraic proof using the kernel method was recently provided by Andrews and Chern, but a bijective proof still remains mysterious. The goal of this article is to establish bijectively both Lin's symmetry conjecture and the $\gamma$-positivity of the ascent polynomial on $\I_n(>,\neq,>)$. The latter result implies that the distribution of ascents on $\I_n(>,\neq,>)$ is symmetric and unimodal.