Quarter-Turn Baxter Permutations

TL;DR

This paper investigates the symmetry properties of Baxter permutations under quarter-turn rotations, revealing a refined enumeration that hints at underlying combinatorial bijections.

Contribution

It introduces a new analysis of Baxter permutations under quarter-turn rotation, extending previous work on half-turn symmetry and employing generating trees for enumeration.

Findings

Number of Baxter permutations fixed under quarter-turn rotation has a refined enumeration.

The enumeration suggests the existence of a new combinatorial bijection.

Refinement of symmetry analysis from half-turn to quarter-turn rotation.

Abstract

Baxter permutations are known to be in bijection with a wide number of combinatorial objects. Previously, it was shown that each of these objects had a natural involution which was carried equivariantly by the known bijections, and the number of objects fixed under involution was given by Stembridge's $q=-1$ phenomenon. In this paper, we consider the order 4 action of a quarter-turn rotation of a Baxter permutation matrix, refining the half-turn rotation previously studied. Using the method of generating trees, we show that the number of Baxter permutations fixed under quarter-turn rotation has a very nice enumeration, which suggests the existence of a combinatorial bijection.