Refined enumeration of permutations sorted with two stacks and a D_8-symmetry

TL;DR

This paper characterizes and enumerates permutations sorted by a composition of stack sorting and D_8-symmetry operations, confirming conjectures and using generating trees and bijections for proofs.

Contribution

It proves conjectures on the enumeration of permutations sorted by operators involving stack sorting and D_8-symmetry, enriching the understanding with classical statistics.

Findings

Confirmed conjectures on permutation enumeration with D_8-symmetry

Characterized permutations via generalized pattern avoidance

Used generating trees and bijections for proofs

Abstract

We study permutations that are sorted by operators of the form $\mathbf{S} \circ \alpha \circ \mathbf{S}$, where $\mathbf{S}$ is the usual stack sorting operator introduced by D. Knuth and $\alpha$ is any $D_8$-symmetry obtained 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 A. Claesson, M. Dukes and E. Steingr\'imsson. 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 S. Giraudo between Baxter and twisted Baxter permutations.