On the set partitions that require maximum sorts through the $aba-$avoiding stack

Abstract

Recently, Xia introduced a deterministic variation $\phi_{\sigma}$ of Defant and Kravitz's stack-sorting maps for set partitions and showed that any set partition $p$ is sorted by $\phi^{N(p)}_{aba}$, where $N(p)$ is the number of distinct alphabets in $p$. Xia then asked which set partitions $p$ are not sorted by $\phi_{aba}^{N(p)-1}$. In this note, we prove that the minimal length of a set partition $p$ that is not sorted by $\phi_{aba}^{N(p)-1}$ is $2N(p)$. Then we show that there is only one set partition of length $2N(p)$ and ${{N(p) + 1} \choose 2} + 2{N(p) \choose 2}$ set partitions of length $2N(p)+1$ that are not sorted by $\phi_{aba}^{N(p)-1}$.