2021/04/06 by Defant, Colin · 1 citation
#05A05 #05E16 #37E15 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2104.02675
Let W be an irreducible Coxeter group. We define the Coxeter pop-stack-sorting operator Pop:W→ W to be the map that fixes the identity element and sends each nonidentity element w to the meet of the elements covered by w in the right weak order. When W is the symmetric group Sn, Pop coincides with the pop-stack-sorting map. Generalizing a theorem about the pop-stack-sorting map due to Ungar, we prove that supw∈ W|OPop(w)|=h, where h is the Coxeter number of W (with h=∞ if W is infinite) and Of(w) denotes the forward orbit of w under a map f. When W is finite, this result is equivalent to the statement that the maximum number of terms appearing in the Brieskorn normal form of an element of W is h-1. More generally, we define a map f:W→ W to be compulsive if for every w∈ W, f(w) is less than or equal to Pop(w) in the right weak order. We prove that if f is compulsive, then supw∈ W|Of(w)|≤ h. This result is new even for symmetric groups. We prove that 2-pop-stack-sortable elements in type B are in bijection with 2-pop-stack-sortable permutations in type A, which were enumerated by Pudwell and Smith. Claesson and Gudmundsson proved that for each fixed nonnegative integer t, the generating function that counts t-pop-stack-sortable permutations in type A is rational; we establish analogous results in types B and \widetilde A.