Counting Pop-Stacked Permutations in Polynomial Time

Claesson, Anders; Guðmundsson, Bjarki Ágúst; Pantone, Jay

doi:10.1080/10586458.2021.1926001

Cited by 7 publications

(9 citation statements)

References 7 publications

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“…This result motivates the study of the sizes of the images of the pop-stack sorting operators on interesting classes of lattices. This investigation was already initiated in [3,13] for the weak order on 𝑆 𝑛 (where Pop ↓ 𝐿 is the classical pop-stack sorting map) and in [43] for the lattice of order ideals of a type A root poset. In Section 11.2, we state several enumerative conjectures about the sizes of the images of pop-stack sorting operators on other specific lattices such as Tamari lattices, bipartite Cambrian lattices and distributive lattices of order ideals in positive root posets.…”

Section: Rowmotion and Pop-stack Sortingmentioning

confidence: 99%

Semidistrim Lattices

Defant

Williams²

2023

Forum of Mathematics, Sigma

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We introduce semidistrim lattices, a simultaneous generalization of semidistributive and trim lattices that preserves many of their common properties. We prove that the elements of a semidistrim lattice correspond to the independent sets in an associated graph called the Galois graph, that products and intervals of semidistrim lattices are semidistrim and that the order complex of a semidistrim lattice is either contractible or homotopy equivalent to a sphere. Semidistrim lattices have a natural rowmotion operator, which simultaneously generalizes Barnard’s $\overline \kappa $ map on semidistributive lattices as well as Thomas and the second author’s rowmotion on trim lattices. Every lattice has an associated pop-stack sorting operator that sends an element x to the meet of the elements covered by x. For semidistrim lattices, we are able to derive several intimate connections between rowmotion and pop-stack sorting, one of which involves independent dominating sets of the Galois graph.

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Section: Rowmotion and Pop-stack Sortingmentioning

confidence: 99%

Semidistrim Lattices

Defant

Williams²

2023

Forum of Mathematics, Sigma

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“…When the additional parameter is the number of runs, this gives a recurrence which encodes the addition of a new run of length k to the end of a given permutation of size n (and relabels this concatenation to get a permutation of size n + k). The cost of this approach is analysed in [14], and was implemented with care, allowing the computation of number of pop-stacked permutations of size n, for n ≥ 1000 with a computer cluster.…”

Section: A Functional Equation For Pop-stacked Permutations and The C...mentioning

confidence: 99%

“…[16, p. 5]). Note that the authors of [14] carried out an experimental analysis using automated fitting and differential approximation. Their analysis of the counting sequence led them to conjecture that the corresponding exponential generating function possesses an infinite number of singularities, thus implying non-D-finiteness.…”

Section: Asymptotics Of Pop-stacked Permutationsmentioning

confidence: 99%

Flip-sort and combinatorial aspects of pop-stack sorting

Asinowski,

Banderier,

Hackl

2020

Preprint

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Flip-sort is a natural sorting procedure which raises fascinating combinatorial questions. It finds its roots in the seminal work of Knuth on stack-based sorting algorithms and leads to many links with permutation patterns. We present several structural, enumerative, and algorithmic results on permutations that need few (resp. many) iterations of this procedure to be sorted. In particular, we give the shape of the permutations after one iteration, and characterize several families of permutations related to the best and worst cases of flip-sort. En passant, we also give some links between pop-stack sorting, automata, and lattice paths, and introduce several tactics of bijective proofs which have their own interest.

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“…This operator is called the pop-stack sorting map because it coincides with the map that passes a permutation through a data structure called a pop-stack in a right-greedy manner. This map has already been studied in combinatorics and theoretical computer science [4,5,23,24,37,39]. If one starts with a permutation in S n and repeatedly applies the pop-stack sorting map, then one will eventually reach the identity permutation 12 • • • n, which is fixed by Pop.…”

Section: Introductionmentioning

confidence: 99%

Ungarian Markov Chains

Defant¹,

Li²

2023

Preprint

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We introduce the Ungarian Markov chain UL associated to a finite lattice L. The states of this Markov chain are the elements of L. When the chain is in a state x ∈ L, it transitions to the meet of {x} ∪ T , where T is a random subset of the set of elements covered by x. We focus on estimating E(L), the expected number of steps of UL needed to get from the top element of L to the bottom element of L. Using direct combinatorial arguments, we provide asymptotic estimates when L is the weak order on the symmetric group Sn and when L is the n-th Tamari lattice. When L is distributive, the Markov chain UL is equivalent to an instance of the well-studied random process known as last-passage percolation with geometric weights. One of our main results states that if L is a trim lattice, then E(L) ≤ E(spine(L)), where spine(L) is a specific distributive sublattice of L called the spine of L. Combining this lattice-theoretic theorem with known results about last-passage percolation yields a powerful method for proving upper bounds for E(L) when L is trim. We apply this method to obtain uniform asymptotic upper bounds for the expected number of steps in the Ungarian Markov chains of Cambrian lattices of classical types and the Ungarian Markov chains of ν-Tamari lattices.

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Counting Pop-Stacked Permutations in Polynomial Time

Cited by 7 publications

References 7 publications

Semidistrim Lattices

Semidistrim Lattices

Flip-sort and combinatorial aspects of pop-stack sorting

Ungarian Markov Chains

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