Sorting and preimages of pattern classes

We introduce an algorithm to determine when a sorting operation, such as stack-sort or bubble-sort, outputs a given pattern. The algorithm provides a new proof of the description of West-2-stack-sortable permutations, that is permutations that are completely sorted when passed twice through a stack, in terms of patterns. We also solve the long-standing problem of describing West-3-stack-sortable permutations. This requires a new type of generalized permutation pattern we call a decorated pattern.Comment: 13 pages, 5 figures, to appear at FPSAC 201

Actions on permutations and unimodality of descent polynomials

We study a group action on permutations due to Foata and Strehl and use it to prove that the descent generating polynomial of certain sets of permutations has a nonnegative expansion in the basis $\{t^i(1+t)^{n-1-2i}\}_{i=0}^m$, $m=\lfloor (n-1)/2 \rfloor$. This property implies symmetry and unimodality. We prove that the action is invariant under stack-sorting which strengthens recent unimodality results of B\'ona. We prove that the generalized permutation patterns $(13-2)$ and $(2-31)$ are invariant under the action and use this to prove unimodality properties for a $q$-analog of the Eulerian numbers recently studied by Corteel, Postnikov, Steingr\'{\i}msson and Williams. We also extend the action to linear extensions of sign-graded posets to give a new proof of the unimodality of the $(P,\omega)$-Eulerian polynomials of sign-graded posets and a combinatorial interpretations (in terms of Stembridge's peak polynomials) of the corresponding coefficients when expanded in the above basis. Finally, we prove that the statistic defined as the number of vertices of even height in the unordered decreasing tree of a permutation has the same distribution as the number of descents on any set of permutations invariant under the action. When restricted to the set of stack-sortable permutations we recover a result of Kreweras.Comment: 19 pages, revised version to appear in Europ. J. Combi

On a Conjecture by Baril, Cerbai, Khalil, and Vajnovszki on Two Restricted Stacks

Let $s$ be West's stack-sorting map, and let $s_{T}$ be the generalized stack-sorting map, where instead of being required to increase, the stack avoids subpermutations that are order-isomorphic to any permutation in the set $T$. In 2020, Cerbai, Claesson, and Ferrari introduced the $\sigma$-machine $s \circ s_{\sigma}$ as a generalization of West's $2$-stack-sorting-map $s \circ s$. As a further generalization, in 2021, Baril, Cerbai, Khalil, and Vajnovski introduced the $(\sigma, \tau)$-machine $s \circ s_{\sigma, \tau}$ and enumerated $|\mathrm{Sort}_{n}(\sigma,\tau)|$ -- the number of permutations in $S_n$ that are mapped to the identity by the $(\sigma, \tau)$-machine -- for six pairs of length $3$ permutations $(\sigma, \tau)$. In this work, we settle a conjecture by Baril, Cerbai, Khalil, and Vajnovski on the only remaining pair of length $3$ patterns $(\sigma, \tau) = (132, 321)$ for which $|\mathrm{Sort}_{n}(\sigma, \tau)|$ appears in the OEIS. In addition, we enumerate $|\mathrm{Sort}_n(123, 321)|$, which does not appear in the OEIS, but has a simple closed form

Pattern Avoidance in Task-Precedence Posets

We have extended classical pattern avoidance to a new structure: multiple task-precedence posets whose Hasse diagrams have three levels, which we will call diamonds. The vertices of each diamond are assigned labels which are compatible with the poset. A corresponding permutation is formed by reading these labels by increasing levels, and then from left to right. We used Sage to form enumerative conjectures for the associated permutations avoiding collections of patterns of length three, which we then proved. We have discovered a bijection between diamonds avoiding 132 and certain generalized Dyck paths. We have also found the generating function for descents, and therefore the number of avoiders, in these permutations for the majority of collections of patterns of length three. An interesting application of this work (and the motivating example) can be found when task-precedence posets represent warehouse package fulfillment by robots, in which case avoidance of both 231 and 321 ensures we never stack two heavier packages on top of a lighter package.Comment: 17 page

Generalized permutation patterns - a short survey

An occurrence of a classical pattern p in a permutation π is a subsequence of π whose letters are in the same relative order (of size) as those in p. In an occurrence of a generalized pattern, some letters of that subsequence may be required to be adjacent in the permutation. Subsets of permutations characterized by the avoidance—or the prescribed number of occurrences— of generalized patterns exhibit connections to an enormous variety of other combinatorial structures, some of them apparently deep. We give a short overview of the state of the art for generalized patterns