Refined Enumeration of Permutations Sorted with Two Stacks and a D8-Symmetry

We study permutations that are sorted by operators of the form S ° α ° S , where S is the usual stack sorting operator introduced by Knuth and α is any D 8 -symmetry obtained by combining the classical reverse, complement, and inverse operations. Such permutations can be characterized by excluded (generalized) patterns. Some conjectures about the enumeration of these permutations, refined with numerous classical statistics, have been proposed by Claesson, Dukes, and Steingrímsson. We prove these conjectures, and enrich one of them with a few more statistics. The proofs mostly rely on generating trees techniques, and on a recent bijection of Giraudo between Baxter and twisted Baxter permutations.