A Note on Flips in Diagonal Rectangulations

TL;DR

This paper explores flip operations on diagonal rectangulations and their correspondence with transpositions in Baxter permutations, providing a complete combinatorial and geometric characterization of these local modifications.

Contribution

It offers a comprehensive analysis of flip operations on diagonal rectangulations and their interpretation as transpositions in Baxter permutations, extending previous work.

Findings

Characterization of flip operations on diagonal rectangulations

Correspondence between flips and transpositions in Baxter permutations

Complete geometric and combinatorial description of flips

Abstract

Rectangulations are partitions of a square into axis-aligned rectangles. A number of results provide bijections between combinatorial equivalence classes of rectangulations and families of pattern-avoiding permutations. Other results deal with local changes involving a single edge of a rectangulation, referred to as flips, edge rotations, or edge pivoting. Such operations induce a graph on equivalence classes of rectangulations, related to so-called flip graphs on triangulations and other families of geometric partitions. In this note, we consider a family of flip operations on the equivalence classes of diagonal rectangulations, and their interpretation as transpositions in the associated Baxter permutations, avoiding the vincular patterns { 3{14}2, 2{41}3 }. This complements results from Law and Reading (JCTA, 2012) and provides a complete characterization of flip operations on diagonal rectangulations, in both geometric and combinatorial terms.