On 021-Avoiding Ascent Sequences

Ascent sequences were introduced by Bousquet-M'{e}lou, Claesson, Dukes and Kitaev in their study of $(\mathbf{2}+\mathbf{2})$-free posets. An ascent sequence of length $n$ is a nonnegative integer sequence $x=x_1{x}_2\mathrm{.}\mathrm{.}\mathrm{.}{x}_n$ such that $x_1=0$ and $x_i\le \text{asc}(x_1{x}_2\mathrm{.}\mathrm{.}\mathrm{.}{x}_{i-1})+1$ for all $1<i\le n$, where $\text{asc}(x_1{x}_2\mathrm{.}\mathrm{.}\mathrm{.}{x}_{i-1})$ is the number of ascents in the sequence $x_1{x}_2\mathrm{.}\mathrm{.}\mathrm{.}{x}_{i-1}$. We let ${\text{cA}}_n$ stand for the set of such sequences and use ${\text{cA}}_n(p)$ 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 ${\text{cA}}_n(021)$ as over $S_n(132)$. Furthermore, they conjectured that the pair $(\text{asc},\text{rlm})$ is equidistributed over ${\text{cA}}_n(021)$ and $S_n(132)$ where $\text{rlm}$ is the right-to-left minima statistic. We prove this conjecture by constructing a bistatistic-preserving bijection.