Permutree sorting

Generalizing stack sorting and c-sorting for permutations, we define the permutree sorting algorithm. Given two disjoint subsets U and D of {2, . . ., n - 1}, the (U, D)-permutree sorting tries to sort the permutation π ∈ Sn and fails if and only if there are 1 ≤ i < j < k ≤ n such that π contains the subword jki if j ∈ U and kij if j ∈ D. This algorithm is seen as a way to explore an automaton which either rejects all reduced words of π, or accepts those reduced words for π whose prefixes are all (U, D)-permutree sortable.