Two Vignettes On Full Rook Placements

Using bijections between pattern-avoiding permutations and certain full rook placements on Ferrers boards, we give short proofs of two enumerative results. The first is a simplified enumeration of the 3124, 1234-avoiding permutations, obtained recently by Callan via a complicated decomposition. The second is a streamlined bijection between 1342-avoiding permutations and permutations which can be sorted by two increasing stacks in series, originally due to Atkinson, Murphy, and Ru\v{s}kuc.Comment: 9 pages, 4 figure

Enumeration of Stack-Sorting Preimages via a Decomposition Lemma

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 ${\bf B}\circ s$, where ${\bf B}$ 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 $s^{-1}(\text{Av}(\tau^{(1)},\ldots,\tau^{(r)}))$ for $\{\tau^{(1)},\ldots,\tau^{(r)}\}\subseteq S_3$ 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.Comment: 20 pages, 4 figures. arXiv admin note: text overlap with arXiv:1903.0913

Generating Permutations with Restricted Containers

We investigate a generalization of stacks that we call $\mathcal{C}$-machines. We show how this viewpoint rapidly leads to functional equations for the classes of permutations that $\mathcal{C}$-machines generate, and how these systems of functional equations can frequently be solved by either the kernel method or, much more easily, by guessing and checking. General results about the rationality, algebraicity, and the existence of Wilfian formulas for some classes generated by $\mathcal{C}$-machines are given. We also draw attention to some relatively small permutation classes which, although we can generate thousands of terms of their enumerations, seem to not have D-finite generating functions

Permutation classes

This is a survey on permutation classes for the upcoming book Handbook of Enumerative Combinatorics

Two vignettes on full rook placements

Using bijections between pattern-avoiding permutations and certain full rook placements on Ferrers boards, we give short proofs of two enumerative results. The first is a simplified enumeration of the 3124, 1234-avoiding permutations, obtained recently by Callan via a complicated decomposition. The second is a streamlined bijection between 1342-avoiding permutations and permutations which can be sorted by two increasing stacks in series, originally due to Atkinson, Murphy, and Ruškuc