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".
This paper continues the analysis of the pattern-avoiding sorting machines recently introduced by Cerbai, Claesson and Ferrari (2020). These devices consist of two stacks, through which a permutation is passed in order to sort it, where the content of each stack must at all times avoid a certain pattern. Here we characterize and enumerate the set of permutations that can be sorted when the first stack is $132$-avoiding, solving one of the open problems proposed by the above mentioned authors. To that end we present several connections with other well known combinatorial objects, such as lattice paths and restricted growth functions (which encode set partitions). We also provide new proofs for the enumeration of some sets of pattern-avoiding restricted growth functions and we expect that the tools introduced can be fruitfully employed to get further similar results.
Stack sorting/two stacks in series Pattern avoiding permutations/machines Catalan and the Schröder numbers Combinatorial problems Pattern avoiding machines were introduced recently by Claesson, Cerbai and Ferrari as a particular case of the two-stacks in series sorting device. They consist of two restricted stacks in series, ruled by a right-greedy procedure and the stacks avoid some specified patterns. Some of the obtained results have been further generalized to Cayley permutations by Cerbai, specialized to particular patterns by Defant and Zheng, or considered in the context of functions over the symmetric group by Berlow. In this work we study pattern avoiding machines where the first stack avoids a pair of patterns of length 3 and investigate those pairs for which sortable permutations are counted by the (binomial transform of the) Catalan numbers and the Schröder numbers.
Pattern avoiding machines were recently introduced by Claesson, Ferrari and the current author to gain a better understanding of the classical 2-stacksort problem. In this paper we generalize these devices by allowing permutations with repeated elements, also known as Cayley permutations. The main result is a description of those patterns such that the corresponding set of sortable permutations is a class. We also show a new involution on the set of Cayley permutations, obtained by regarding a pattern-avoiding stack as an operator. Finally, we analyze two generalizations of pop-stack sorting on Cayley permutations. In both cases we describe sortable permutations in terms of pattern avoidance. * The author is member of the INdAM Research group GNCS; he is partially supported by INdAM -GNCS 2019 project "Proprietá combinatorie e rilevamento di pattern in strutture discrete lineari e bidimensionali".
We take the first steps in developing a theory of transport of patterns from Fishburn permutations to (modified) ascent sequences. Given a set of pattern avoiding Fishburn permutations, we provide an explicit construction for the basis of the corresponding set of modified ascent sequences. Our approach is in fact more general and can transport patterns between permutations and equivalence classes of so called Cayley permutations. This transport of patterns relies on a simple operation we call the Burge transpose. It operates on certain biwords called Burge words. Moreover, using mesh patterns on Cayley permutations, we present an alternative view of the transport of patterns as a Wilf-equivalence between subsets of Cayley permutations. We also highlight a connection with primitive ascent sequences.
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