A Survey of Stack-Sorting Disciplines

Bóna, Miklós

doi:10.37236/1693

Cited by 87 publications

(106 citation statements)

References 32 publications

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“…Recall that a sequence (a n ) n≥1 of positive numbers is called log-convex if (a n+1 /a n ) n≥1 is nondecreasing. 3,8]). For all n, t ≥ 1, we have W t (n) ≤ (t + 1)n n .…”

Section: Theorem 13 ([14]mentioning

confidence: 99%

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Counting 3-stack-sortable permutations

Defant

2020

Journal of Combinatorial Theory, Series A

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We prove a "decomposition lemma" that allows us to count preimages of certain sets of permutations under West's stack-sorting map s. As a first application, we give a new proof of Zeilberger's formula for the number W2(n) of 2-stack-sortable permutations in Sn. Our proof generalizes, allowing us to find an algebraic equation satisfied by the generating function that counts 2-stack-sortable permutations according to length, number of descents, and number of peaks. This is also the first proof of this formula that generalizes to the setting of 3-stack-sortable permutations. Indeed, the same method allows us to obtain a recurrence relation for W3(n), the number of 3-stacksortable permutations in Sn. Hence, we obtain the first polynomial-time algorithm for computing these numbers. We compute W3(n) for n ≤ 174, vastly extending the 13 terms of this sequence that were known before. We also prove the first nontrivial lower bound for lim n→∞ W3(n) 1/n , showing that it is at least 8.659702. Invoking a result of Kremer, we also prove that lim n→∞ Wt(n) 1/n ≥ ( √ t + 1) 2 for all t ≥ 1, which we use to improve a result of Smith concerning a variant of the stack-sorting procedure. Our computations allow us to disprove a conjecture of Bóna, although we do not yet know for sure which one.In fact, we can refine our methods to obtain a recurrence for W3(n, k, p), the number of 3-stacksortable permutations in Sn with k descents and p peaks. This allows us to gain a large amount of evidence supporting a real-rootedness conjecture of Bóna. Using part of the theory of valid hook configurations, we give a new proof of a γ-nonnegativity result of Brändén, which in turn implies an older result of Bóna. We then answer a question of the current author by producing a set A ⊆ S11 such that σ∈s −1 (A) x des(σ) has nonreal roots. We interpret this as partial evidence against the same real-rootedness conjecture of Bóna that we found evidence supporting. Examining the parities of the numbers W3(n), we obtain strong evidence against yet another conjecture of Bóna. We end with some conjectures of our own.

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“…Recall that a sequence (a n ) n≥1 of positive numbers is called log-convex if (a n+1 /a n ) n≥1 is nondecreasing. 3,8]). For all n, t ≥ 1, we have W t (n) ≤ (t + 1)n n .…”

Section: Theorem 13 ([14]mentioning

confidence: 99%

“…We have suppressed the dependence on x and y in our notation for readability. Let L r (x, y) be as in (8).…”

Section: Now Let Bmentioning

confidence: 99%

Counting 3-stack-sortable permutations

Defant

2020

Journal of Combinatorial Theory, Series A

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“…On the left is the plot of 2 7 3 5 9 4 8 1 6 10 11 12 along with one hook whose southwest endpoint is (5,9) and whose northeast endpoint is (11,11). The points lying strictly below this hook are (6, 4), (7,8), (8,1), (9,6), (10,10). These five points and (11,11) are the points lying weakly below the hook.…”

Section: Stack-sorting Backgroundmentioning

confidence: 99%

“…This map, denoted by s, sends permutations of length n to permutations of length n. It is a slight variant of the stack-sorting algorithm that Knuth introduced in [27]. The map s was studied extensively in West's 1990 Ph.D. thesis [44] and has received a considerable amount of attention ever since [5,6,8,14]. We give necessary background results concerning the stack-sorting map in Section 3, but the reader seeking additional historical motivation should consult [5,6,14] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Catalan intervals and uniquely sorted permutations

Defant

2020

Journal of Combinatorial Theory, Series A

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“…• Sorting integer values with the help of stacks or queues is of some importance in informatics and also closely related to very general mathematical problems that extract subsequences from integer sequences. Instead of trying to summarize the rich literature on this topic, we refer to the survey paper of Bona [3]. This field rather takes a reverted view on the problem (e.g.…”

Section: Literature Reviewmentioning

confidence: 99%

Resequencing with parallel queues to minimize the maximum number of items in the overflow area

Boywitz

Boysen

Briskorn

2016

Naval Research Logistics

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This article treats an elementary optimization problem, where an inbound stream of successive items is to be resequenced with the help of multiple parallel queues in order to restore an intended target sequence. Whenever early items block the one item to be currently released into the target sequence, they are withdrawn from their queue and intermediately stored in an overflow area until their actual release is reached. We aim to minimize the maximum number of items simultaneously stored in the overflow area during the complete resequencing process. We met this problem in industry practice at a large German automobile producer, who has to resequence containers with car seats prior to the assembly process. We formalize the resulting resequencing problem and provide suited exact and heuristic solution algorithms. In our computational study, we also address managerial aspects such as how to properly avoid the negative effects of sequence alterations. © 2016 Wiley Periodicals, Inc. Naval Research Logistics 63: 401–415, 2016

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A Survey of Stack-Sorting Disciplines

Cited by 87 publications

References 32 publications

Counting 3-stack-sortable permutations

Counting 3-stack-sortable permutations

Catalan intervals and uniquely sorted permutations

Resequencing with parallel queues to minimize the maximum number of items in the overflow area

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