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The Hopf algebra of diagonal rectangulations
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 sub
Hopf algebra 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.Comment: Very minor changes from version 1, in response to comments by
referees. This is the final version, to appear in JCTA. 43 pages, 17 figure
Baxter permutations rise again
AbstractBaxter permutations, so named by Boyce, were introduced by Baxter in his study of the fixed points of continuous functions which commute under composition. Recently Chung, Graham, Hoggatt, and Kleiman obtained a sum formula for the number of Baxter permutations of 2n − 1 objects, but admit to having no interpretation of the individual terms of this sum. We show that in fact the kth term of this sum counts the number of (reduced) Baxter permutations that have exactly k − 1 rises
Bijections for Baxter Families and Related Objects
The Baxter number can be written as . These
numbers have first appeared in the enumeration of so-called Baxter
permutations; is the number of Baxter permutations of size , and
is the number of Baxter permutations with descents and
rises. With a series of bijections we identify several families of
combinatorial objects counted by the numbers . Apart from Baxter
permutations, these include plane bipolar orientations with vertices and
faces, 2-orientations of planar quadrangulations with white and
black vertices, certain pairs of binary trees with left and
right leaves, and a family of triples of non-intersecting lattice paths. This
last family allows us to determine the value of as an
application of the lemma of Gessel and Viennot. The approach also allows us to
count certain other subfamilies, e.g., alternating Baxter permutations, objects
with symmetries and, via a bijection with a class of plan bipolar orientations
also Schnyder woods of triangulations, which are known to be in bijection with
3-orientations.Comment: 31 pages, 22 figures, submitted to JCT
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