Enumeration of Stack-Sorting Preimages via a Decomposition Lemma

Abstract

We give three applications of a recently-proven "Decomposition Lemma," which allows one to count preimages of certain sets of permutations under West's stack-sorting map $s$. We first enumerate the permutation class $s^{-1}(\text{Av}(231,321))=\text{Av}(2341,3241,45231)$, finding a new example of an unbalanced Wilf equivalence. This result is equivalent to the enumeration of permutations sortable by $\mathbf{B}\circ s$, where $\mathbf{B}$ is the bubble sort map. We then prove that the sets $s^{-1}(\text{Av}(231,312))$, $s^{-1}(\text{Av}(132,231))=\text{Av}(2341,1342,\underline{32}41,\underline{31}42)$, and $s^{-1}(\text{Av}(132,312))=\text{Av}(1342,3142,3412,34\underline{21})$ are counted by the so-called "Boolean-Catalan numbers," settling a conjecture of the current author and another conjecture of Hossain. This completes the enumerations of all sets of the form $s^{-1}(\text{Av}(\ddot{I}{„}^{(1)},\dots ,\ddot{I}{„}^{(r)}))$ for $\{\ddot{I}{„}^{(1)},\dots ,\ddot{I}{„}^{(r)}\}\subseteq S_3$ with the exception of the set $\{321\}$. We also find an explicit formula for $|s^{-1}({\text{Av}}_{n,k}(231,312,321))|$, where ${\text{Av}}_{n,k}(231,312,321)$ is the set of permutations in ${\text{Av}}_n(231,312,321)$ with $k$ descents. This allows us to prove a conjectured identity involving Catalan numbers and order ideals in Young's lattice.

20 pages, 4 figures. arXiv admin note: text overlap with arXiv:1903.09138