A Note on Flips in Diagonal Rectangulations

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.

References in corpus (7)

• Bijections for Baxter Families and Related Objects
• Algebraic and combinatorial structures on pairs of twin binary trees
• The Hopf algebra of diagonal rectangulations
• Cambrian Hopf Algebras
• Generic rectangulations
• A History of Flips in Combinatorial Triangulations
• The Flip Diameter of Rectangulations and Convex Subdivisions

Cited by in corpus (1)

• Rectangulotopes