Stack-sorting for Coxeter groups

Abstract

Given an essential semilattice congruence ≡ on the left weak order of a Coxeter group W , we define the Coxeter stack-sorting operator S ≡ : W → W by S ≡ (w) = w π ≡ ↓ (w) −1 , where π ≡ ↓ (w) is the unique minimal element of the congruence class of ≡ containing w. When ≡ is the sylvester congruence on the symmetric group S n , the operator S ≡ is West's stack-sorting map. When ≡ is the descent congruence on S n , the operator S ≡ is the pop-stack-sorting map. We establish several general results about Coxeter stack-sorting operators, especially those acting on symmetric groups. For example, we prove that if ≡ is an essential lattice congruence on S n , then every permutation in the image of S ≡ has at most 2(n−1) /3 right descents; we also show that this bound is tight.We then introduce analogues of permutree congruences in types B and A and use them to isolate Coxeter stack-sorting operators s B and s that serve as canonical type-B and type-A counterparts of West's stack-sorting map. We prove analogues of many known results about West's stack-sorting map for the new operators s B and s. For example, in type A, we obtain an analogue of Zeilberger's classical formula for the number of 2-stack-sortable permutations in S n .

Coxeter Pop-Tsack Torsing

“…Indeed, as discussed in [19], this operator agrees with the pop-stack sorting map when W is the symmetric group S n . Observe that Pop S fixes the identity element e ∈ W .…”

“…This map, which originally appeared in an article by Ungar concerning directions determined by points in the plane [28], has gained interest in the past few years among enumerative combinatorialists [1,5,4,20,17,18,22]. See [19] and the references therein for several other dynamical systems arising from sorting operators on permutations.…”

“…One of the main results from [19] is the following generalization of Ungar's theorem to arbitrary finite irreducible Coxeter groups. Let us also remark that, by construction, the elements of W that get sorted into the identity e in at most one iteration of Pop S are exactly the longest elements of standard parabolic subgroups-hence, there are 2 |S| such elements.…”

“…[19]). Let W be a finite irreducible Coxeter group with a set S of simple reflections and with Coxeter number h. Then max w∈W O Pop S (w) = h.…”

The Pop-stack-sorting Operator on Tamari Lattices

“…Claesson and Guðmundsson [7] proved that for each fixed nonnegative integer t, the generating function that counts t-Pop-sortable elements in S n is rational. Defant [9] established the analogous rationality result for the generating functions of t-Pop-sortable elements of type B and type A weak orders.…”