Stack-sorting for Coxeter groups

Given an essential semilattice congruence \(\equiv\) on the left weak order of a \linebreak Coxeter group \(W\), we define stack-sorting operator \({\bf S}_\equiv:W\to W\) by \({{\bf S}_\equiv(w)=w\big(\pi_\downarrow^\equiv(w)\big)^{-1}}\), where \(\pi_\downarrow^\equiv(w)\) is unique minimal element class containing \(w\). When sylvester symmetric \(S_n\), S}_\equiv\) West's map. descent pop-stack-sorting We establish several general results about operators, especially those acting groups. For example, prove that if lattice then every permutation in image has at most \(\left\lfloor\frac{2(n-1)}{3}\right\rfloor\) right descents; also show this bound tight. introduce analogues permutree congruences types \(B\) and \(\widetilde A\) use them to isolate operators \(\mathtt{s}_B\) \(\widetilde{\mathtt{s}}\) serve as canonical type-\(B\) type-\(\widetilde counterparts many known map for new \(\widetilde{\mathtt{s}}\). type A\), obtain analogue Zeilberger's classical formula number \(2\)-stack-sortable permutations \(S_n\).Mathematics Subject Classifications: 06A12, 06B10, 37E15, 05A05, 05E16