Comparing Algorithms for Sorting with t Stacks in Series

Smith, Rebecca

doi:10.1007/s00026-004-0209-3

Cited by 6 publications

(5 citation statements)

References 4 publications

(2 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In Section 7, we prove that lim n→∞ W t (n) 1/n ≥ ( √ t + 1) 2 for every t ≥ 1, yielding the first nontrivial lower bounds for these growth rates for all t ≥ 4. As a corollary, we improve a result of Smith concerning permutations that can be sorted by t stacks in series using the so-called "left-greedy algorithm" [43]. Although there are multiple ways one could rigorously interpret Bóna's Conjecture 1.3, we will see in Section 6 that every reasonable interpretation of the conjecture is likely to be false.…”

Section: Conjecture 14 ([9]mentioning

confidence: 87%

“…This completes the induction and proves the following theorem. In [43], Smith investigated a variant of the stack-sorting map known as the "left-greedy algorithm." Let W t (n) be the set of permutations in S n that can be sorted by t stacks in series using the left-greedy algorithm (see her paper for definitions).…”

Section: Lower Bounds For T-stack-sortable Permutationsmentioning

confidence: 99%

See 1 more Smart Citation

Counting 3-stack-sortable permutations

Defant

2020

Journal of Combinatorial Theory, Series A

View full text Add to dashboard Cite

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.

show abstract

Section: Conjecture 14 ([9]mentioning

confidence: 87%

Section: Lower Bounds For T-stack-sortable Permutationsmentioning

confidence: 99%

Counting 3-stack-sortable permutations

Defant

2020

Journal of Combinatorial Theory, Series A

View full text Add to dashboard Cite

show abstract

“…. , 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.…”

Section: Proofmentioning

confidence: 99%

“…Setting Sort (lg) k = {π : π is sorted by the left-greedy procedure}, it turns out that Sort (lg) k is in fact a class which we are able to characterize completely. The choice of a left-greedy strategy, instead of a right-greedy one, is suggested by the results contained in [9]. Proof.…”

Section: A Left-greedy Algorithmmentioning

confidence: 99%

Stack Sorting with Increasing and Decreasing Stacks

Cerbai

Cioni

Ferrari

2020

Electron. J. Combin.

View full text Add to dashboard Cite

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".

show abstract

“…Of course other stack sorting algorithms are possible, and for more than one stack, the iterated greedy algorithm described above is not the optimal algorithm (see, for example, Smith [19]). By allowing any sorting algorithm we reach the definition of general t-stack sortability.…”

Section: Stack Sortingmentioning

confidence: 99%

Problems and conjectures

Vatter¹

2010

Permutation Patterns

View full text Add to dashboard Cite

Comparing Algorithms for Sorting with t Stacks in Series

Cited by 6 publications

References 4 publications

Counting 3-stack-sortable permutations

Counting 3-stack-sortable permutations

Stack Sorting with Increasing and Decreasing Stacks

Problems and conjectures

Contact Info

Product

Resources

About