Pop-stack-sorting for Coxeter groups

Let \(W\) be an irreducible Coxeter group. We define the pop-stack-sorting operator \(\mathsf{Pop}:W\to W\) to map that fixes identity element and sends each nonidentity \(w\) meet of elements covered by in right weak order. When is symmetric group \(S_n\), \(\mathsf{Pop}\) coincides with map. Generalizing a theorem about due Ungar, we prove \[\sup\limits_{w\in W}\left|O_{\mathsf{Pop}}(w)\right|=h,\] where \(h\) number (with \(h=\infty\) if infinite) \(O_f(w)\) denotes forward orbit under \(f\). finite, this result equivalent statement maximum terms appearing Brieskorn normal form \(h-1\). More generally, \(f:W\to compulsive for every \(w\in W\), \(f(w)\) less than or equal \(\mathsf{Pop}(w)\) \(f\) compulsive, then \(\sup\limits_{w\in W}|O_f(w)|\leq h\). This new even groups. \(2\)-pop-stack-sortable type \(B\) are bijection permutations \(A\), which were enumerated Pudwell Smith. Claesson Gu{\dh}mundsson proved fixed nonnegative integer \(t\), generating function counts \(t\)-pop-stack-sortable \(A\) rational; establish analogous results types \(\widetilde A\).Mathematics Subject Classifications: 05E16, 37E15, 05A05Keywords: Pop-stack-sorting, group, order, number, map, regular language