Stack sorting with restricted stacks

Abstract

The (classical) problem of characterizing and enumerating permutations that can be sorted using two stacks connected in series is still largely open. In the present paper we address a related problem, in which we impose restrictions both on the procedure and on the stacks. More precisely, we consider a greedy algorithm where we perform the rightmost legal operation (here "rightmost" refers to the usual representation of stack sorting problems). Moreover, the first stack is required to be σ-avoiding, for some permutation σ, meaning that, at each step, the elements maintained in the stack avoid the pattern σ when read from top to bottom. Since the set of permutations which can be sorted by such a device (which we call σ-machine) is not always a class, it would be interesting to understand when it happens. We will prove that the set of σ-machines whose associated sortable permutations are not a class is counted by Catalan numbers. Moreover, we will analyze two specific σ-machines in full details (namely when σ = 321 and σ = 123), providing for each of them a complete characterization and enumeration of sortable permutations. * G.C. and L.F. are members of the INdAM Research group GNCS; they are partially supported by INdAM -GNCS 2019 project "Studio di proprietá combinatoriche di linguaggi formali ispirate dalla biologia e da strutture bidimensionali" and by a grant of the "Fondazione della Cassa di Risparmio di Firenze" for the project "Rilevamento di pattern: applicazioni a memorizzazione basata sul DNA, evoluzione del genoma, scelta sociale".

Stack Sorting with Increasing and Decreasing Stacks

“…. , 6,9,4,7. Then the set of permutations {α (j) } j≥0 constitutes an infinite antichain in the permutation pattern poset, each of whose element is not 2-sortable.…”

Catalan and Schröder permutations sortable by two restricted stacks

“…Pattern avoiding machines were recently introduced in [7] in attempt to gain a better understanding of sortable permutations using stacks in series. They consist of two restricted stacks in series, equipped with a right-greedy procedure, where the first stack avoids a fixed pattern, reading the elements from top to bottom; and the second stack avoids the pattern 21 (which is a necessary condition for the machine to sort permutations).…”

“…They consist of two restricted stacks in series, equipped with a right-greedy procedure, where the first stack avoids a fixed pattern, reading the elements from top to bottom; and the second stack avoids the pattern 21 (which is a necessary condition for the machine to sort permutations). The authors of [7] provide a characterization of the avoided patterns for which sortable permutations do not form a class, and they show that those patterns are enumerated by the Catalan numbers. For specific patterns, such as 123 and the decreasing pattern of any length, a geometrical description of sortable permutations is also obtained.…”

“…In this work we study a variant of pattern-avoiding machines where the first stack avoids (σ , τ ), a pair of patterns of length three. Following [7], we call it (σ , τ )machine. More specifically, we restrict ourselves to those pairs of patterns for which sortable permutations are counted by either the Catalan numbers or two of their close relatives: the binomial transform of Catalan numbers and the Schröder numbers.…”

L-convex polyominoes and 201-avoiding ascent sequences

“…The connection between pattern-avoiding ascent sequences and other combinatorial objects, such as set partitions, is the subject of [5], while the connection between pattern-avoiding ascent sequences and a number of stack sorting problems is explored in [4].…”