The Hopf algebra of diagonal rectangulations

Authors:
Shirley Law;Nathan Reading
Affiliations:
Department of Mathematics, North Carolina State University, Raleigh, NC, USA;Department of Mathematics, North Carolina State University, Raleigh, NC, USA
Venue:
Journal of Combinatorial Theory Series A
Year:
2012

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Abstract

We define and study a combinatorial Hopf algebra dRec with basis elements indexed by diagonal rectangulations of a square. This Hopf algebra provides an intrinsic combinatorial realization of the Hopf algebra tBax of twisted Baxter permutations, which previously had only been described extrinsically as a Hopf subalgebra of the Malvenuto-Reutenauer Hopf algebra of permutations. We describe the natural lattice structure on diagonal rectangulations, analogous to the Tamari lattice on triangulations, and observe that diagonal rectangulations index the vertices of a polytope analogous to the associahedron. We give an explicit bijection between twisted Baxter permutations and the better-known Baxter permutations, and describe the resulting Hopf algebra structure on Baxter permutations.