vol. 22 no. 2, Permutation Patterns 2019


1. Bounded affine permutations I. Pattern avoidance and enumeration

Neal Madras ; Justin M. Troyka.
We introduce a new boundedness condition for affine permutations, motivated by the fruitful concept of periodic boundary conditions in statistical physics. We study pattern avoidance in bounded affine permutations. In particular, we show that if $\tau$ is one of the finite increasing oscillations, then every $\tau$-avoiding affine permutation satisfies the boundedness condition. We also explore the enumeration of pattern-avoiding affine permutations that can be decomposed into blocks, using analytic methods to relate their exact and asymptotic enumeration to that of the underlying ordinary permutations. Finally, we perform exact and asymptotic enumeration of the set of all bounded affine permutations of size $n$. A companion paper will focus on avoidance of monotone decreasing patterns in bounded affine permutations.

2. Enumeration of Permutation Classes and Weighted Labelled Independent Sets

Christian Bean ; Émile Nadeau ; Henning Ulfarsson.
In this paper, we study the staircase encoding of permutations, which maps a permutation to a staircase grid with cells filled with permutations. We consider many cases, where restricted to a permutation class, the staircase encoding becomes a bijection to its image. We describe the image of those restrictions using independent sets of graphs weighted with permutations. We derive the generating function for the independent sets and then for their weighted counterparts. The bijections we establish provide the enumeration of permutation classes. We use our results to uncover some unbalanced Wilf-equivalences of permutation classes and outline how to do random sampling in the permutation classes. In particular, we cover the classes $\mathrm{Av}(2314,3124)$, $\mathrm{Av}(2413,3142)$, $\mathrm{Av}(2413,3124)$, $\mathrm{Av}(2413,2134)$ and $\mathrm{Av}(2314,2143)$, as well as many subclasses.

3. Enumeration of Stack-Sorting Preimages via a Decomposition Lemma

Colin Defant.
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.

4. Catalan words avoiding pairs of length three patterns

Jean-Luc Baril ; Carine Khalil ; Vincent Vajnovszki.
Catalan words are particular growth-restricted words counted by the eponymous integer sequence. In this article we consider Catalan words avoiding a pair of patterns of length 3, pursuing the recent initiating work of the first and last authors and of S. Kirgizov where (among other things) the enumeration of Catalan words avoiding a patterns of length 3 is completed. More precisely, we explore systematically the structural properties of the sets of words under consideration and give enumerating results by means of recursive decomposition, constructive bijections or bivariate generating functions with respect to the length and descent number. Some of the obtained enumerating sequences are known, and thus the corresponding results establish new combinatorial interpretations for them.

5. Flip-sort and combinatorial aspects of pop-stack sorting

Andrei Asinowski ; Cyril Banderier ; Benjamin Hackl.
Flip-sort is a natural sorting procedure which raises fascinating combinatorial questions. It finds its roots in the seminal work of Knuth on stack-based sorting algorithms and leads to many links with permutation patterns. We present several structural, enumerative, and algorithmic results on permutations that need few (resp. many) iterations of this procedure to be sorted. In particular, we give the shape of the permutations after one iteration, and characterize several families of permutations related to the best and worst cases of flip-sort. En passant, we also give some links between pop-stack sorting, automata, and lattice paths, and introduce several tactics of bijective proofs which have their own interest.

6. Fillings of skew shapes avoiding diagonal patterns

Vít Jelínek ; Mark Karpilovskij.
A skew shape is the difference of two top-left justified Ferrers shapes sharing the same top-left corner. We study integer fillings of skew shapes. As our first main result, we show that for a specific hereditary class of skew shapes, which we call D-free shapes, the fillings that avoid a north-east chain of size $k$ are in bijection with fillings that avoid a south-east chain of the same size. Since Ferrers shapes are a subclass of D-free shapes, this result can be seen as a generalization of previous analogous results for Ferrers shapes. As our second main result, we construct a bijection between 01-fillings of an arbitrary skew shape that avoid a south-east chain of size 2, and the 01-fillings of the same shape that simultaneously avoid a north-east chain of size 2 and a particular non-square subfilling. This generalizes a previous result for transversal fillings.