Pop-stack-sorting for Coxeter groups

Defant, Colin

doi:10.5070/c62359167

Cited by 4 publications

(4 citation statements)

References 52 publications

(76 reference statements)

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“…As mentioned in the introduction, coincides with the classical pop-stack sorting map (see [12, 21]) when L is the right weak order on the symmetric group . Pop-stack sorting operators on lattices were introduced in [15, 14] as generalizations of the pop-stack sorting map.…”

Section: Pop-stack Sorting and Rowmotionmentioning

confidence: 99%

“…Recently, the pop-stack sorting map has received significant attention by combinatorialists [4, 3, 21, 12, 36]. The first author has previously studied pop-stack sorting operators on weak orders of arbitrary Coxeter groups in [15] and on -Tamari lattices in [14]. Mühle [34] studied when L is congruence-uniform, where he called the nucleus of x .…”

Section: Introductionmentioning

confidence: 99%

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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: Pop-stack Sorting and Rowmotionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Semidistrim Lattices

Defant

Williams²

2023

Forum of Mathematics, Sigma

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“…Let L be a lattice. In [28,29], the first author defined the pop-stack sorting operator Pop L : L → L to be the map that acts on an element of L by applying a maximal Ungar move; that is,…”

Section: Introductionmentioning

confidence: 99%

“…The articles [22,28,29,32,34,47] study the dynamical properties and the image of Pop L for various choices of interesting lattices L.…”

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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