On 021-Avoiding Ascent Sequences

Abstract

Ascent sequences were introduced by Bousquet-Mélou, Claesson, Dukes and Kitaev in their study of $(2+2)$-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 $1<i\le n$, 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\'ımsson 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.

6 pages