Sorted and/or sortable permutations

Abstract

In his Ph.D. thesis, Julian West (Permutations with restricted subsequences and stack-sortable permutations, MIT, 1990) studied in depth a map Π that acts on permutations of the symmetric group S n by partially sorting them through a stack. The main motivation of this paper is to characterize and count the permutations of Π( S n) , which we call sorted permutations. This is equivalent to counting preorders of increasing binary trees. We first find a local characterization of sorted permutations. Then, using an extension of Zeilberger's factorization of two-stack sortable permutations (D. Zeilberger, Discrete Math. 102 (1992) 85–93), we obtain for the generating function of sorted permutations an unusual functional equation. Out of curiosity, we apply the same treatment to four other families of permutations (general permutations, one-stack sortable permutations, two-stack sortable permutations, sorted and sortable permutations) and compare the functional equations we obtain. All of them have similar features, involving a divided difference. Moreover, most of them have interesting q-analogs obtained by counting inversions. We solve (some of) our equations.