On 021-Avoiding Ascent Sequences

TL;DR

This paper studies 021-avoiding ascent sequences, proving a conjecture that the ascent and right-to-left minima statistics are equidistributed with 132-avoiding permutations, using a bijection that preserves these statistics.

Contribution

The paper proves Duncan and Steingrímsson's conjecture by constructing a bijection that preserves ascent and right-to-left minima statistics between specific ascent sequences and permutations.

Findings

The ascent statistic is equidistributed over 1-avoiding sequences and 132-avoiding permutations.

A bijection preserving ascent and right-to-left minima statistics is constructed.

The conjecture by Duncan and Steingrímsson is confirmed.

Abstract

Ascent sequences were introduced by Bousquet-M\'{e}lou, Claesson, Dukes and Kitaev in their study of $(\bf{2+2})$-free posets. An ascent sequence of length $n$ is a nonnegative integer sequence $x=x_{1}x_{2}... x_{n}$ such that $x_{1}=0$ and $x_{i}\leq \asc(x_{1}x_{2}...x_{i-1})+1$ for all $1<i\leq n$, where $\asc(x_{1}x_{2}...x_{i-1})$ is the number of ascents in the sequence $x_{1}x_{2}... x_{i-1}$. We let $\cA_n$ stand for the set of such sequences and use $\cA_n(p)$ for the subset of sequences avoiding a pattern $p$. Similarly, we let $S_{n}(\tau)$ be the set of $\tau$-avoiding permutations in the symmetric group $S_{n}$. Duncan and Steingr\'{\i}msson have shown that the ascent statistic has the same distribution over $\cA_n(021)$ as over $S_n(132)$. Furthermore, they conjectured that the pair $(\asc, \rlm)$ is equidistributed over $\cA_n(021)$ and $S_n(132)$ where $\rlm$ is the right-to-left minima statistic. We prove this conjecture by constructing a bistatistic-preserving bijection.