Two Stacks in Series: A Decreasing Stack Followed by an Increasing Stack

Smith, Rebecca

doi:10.1007/s00026-014-0227-8

Cited by 13 publications

(24 citation statements)

References 5 publications

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“…If σ = 12, we have the following result, which completely characterize and enumerate 12-sortable permutations. We remark that the 12-machine is different from the one considered in [9], which is constituted by a decreasing stack and an increasing stack connected in series.…”

Section: Preliminaries and Notationsmentioning

confidence: 89%

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Stack sorting with restricted stacks

Cerbai

Claesson

Ferrari

2020

Journal of Combinatorial Theory, Series A

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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".

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Section: Preliminaries and Notationsmentioning

confidence: 89%

“…Another possible variation on the two-stacks problem is to impose some restrictions on the content of the stack. Rebecca Smith [9] has studied the case in which the first stack is required to be decreasing. Notice that, if we do not choose a specific algorithm in advance, the second stack turns out to be necessarily increasing.…”

Section: Introductionmentioning

confidence: 99%

Stack sorting with restricted stacks

Cerbai

Claesson

Ferrari

2020

Journal of Combinatorial Theory, Series A

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“…• study the machine consisting of two passes through the DI machine described in [10]: are there analogies with West 2-stack-sortable permutations?…”

Section: Final Remarksmentioning

confidence: 99%

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confidence: 99%

Stack Sorting with Increasing and Decreasing Stacks

Cerbai

Cioni

Ferrari

2020

Electron. J. Combin.

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We introduce a sorting machine consisting of k + 1 stacks in series: the first k stacks can only contain elements in decreasing order from top to bottom, while the last one has the opposite restriction. This device generalizes [10], which studies the case k = 1. Here we show that, for k = 2, the set of sortable permutations is a class with infinite basis, by explicitly finding an antichain of minimal nonsortable permutations. This construction can easily be adapted to each k ≥ 3. Next we describe an optimal sorting algorithm, again for the case k = 2. We then analyze two types of left-greedy sorting procedures, obtaining complete results in one case and only some partial results in the other one. We close the paper by discussing a few open questions. * G. C. and L. F. are members of the INdAM Research group GNCS; they are partially supported by IN-dAM -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".

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“…Knuth characterised the set of permutations that can be sorted by a single pass through an infinite stack as the set of permutations that avoid 231 [11]. Since then many variants of the problem have been studied, for example [1,2,3,4,5,6,7,8,9,13,14,15,16,17,18]. The set of permutations sortable by a stack of depth 2 and an infinite stack in series has a basis of 20 permutations [7], while for two infinite stacks in series there is no finite basis [12].…”

Section: Introductionmentioning

confidence: 99%