Algebraic and combinatorial structures on pairs of twin binary trees

Giraudo, Samuele

doi:10.1016/j.jalgebra.2012.03.020

Cited by 62 publications

(80 citation statements)

References 27 publications

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“…Each congruence class is given by a box on the left and corresponds to a binary tree on the right. the permutree lattices [15], the increasing flip lattice on acyclic twists [14], the rotation lattice on diagonal rectangulations [6,10], etc.…”

Section: Introductionmentioning

confidence: 99%

Quotientopes

Pilaud

Santos

2019

Bull. London Math. Soc.

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Section: Introductionmentioning

confidence: 99%

Quotientopes

Pilaud

Santos

2019

Bull. London Math. Soc.

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“…The Baxter congruence (see [11]), denoted by ≡ Bx , is the reflexive and transitive closure of the Baxter adjacency relation ←→ Bx defined for u, v ∈ A * and a, b, c, d ∈ A by…”

Section: Computer Experimentsmentioning

confidence: 99%

“…We conclude this section by providing a method to construct Hopf subalgebras of PM k , analogous to the construction of Hopf subalgebras of FQSym by good congruences [10,14]. The analogs of the Sylvester [15,16], plactic [22,25], hypoplactic [19,20], Bell [32], and Baxter [11] congruences are still good congruences in our context and give rise to Hopf subalgebras of PM k . We end this article by Section 4 where we show that PMN 1 contains a Hopf subalgebra whose bases are indexed by alternating sign matrices, denoted by ASM.…”

Section: Introductionmentioning

confidence: 96%

Hopf algebra structure on packed square matrices

Cheballah

Giraudo

Maurice

2015

Journal of Combinatorial Theory, Series A

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International audienceWe construct a new bigraded Hopf algebra whose bases are indexed by square matrices with entries in the alphabet {0, 1,. .. , k}, k 1, without null rows or columns. This Hopf algebra generalizes the one of permutations of Malvenuto and Reutenauer, the one of k-colored permutations of Novelli and Thibon, and the one of uniform block permutations of Aguiar and Orellana. We study the algebraic structure of our Hopf algebra and show, by exhibiting multiplicative bases, that it is free. We moreover show that it is self-dual and admits a bidendriform bialgebra structure. Besides, as a Hopf subalgebra, we obtain a new one indexed by alternating sign matrices. We study some of its properties and algebraic quotients defined through alternating sign matrices statistics

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“…They were first introduced in [3]. It was shown in [8] that the set of Baxter permutations has one-to-one correspondences to many interesting objects in the so-called Baxter combinatorial family. For examples, [4] showed that plane bipolar orientations with n edges have a one-to-one correspondence with Baxter permutations of length n. [5] establishes a relationship between Baxter permutations and pairs of alternating sign matrices.…”

Section: Baxter Permutationsmentioning

confidence: 99%

A simple optimal binary representation of mosaic floorplans and Baxter permutations

2014

Theoretical Computer Science

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A floorplan is a rectangle subdivided into smaller rectangular sections by horizontal and vertical line segments. Each section in the floorplan is called a block. Two floorplans are considered equivalent if and only if there is a one-to-one correspondence between the blocks in the two floorplans such that the relative position relationship of the blocks in one floorplan is the same as the relative position relationship of the corresponding blocks in another floorplan. The objects of Mosaic floorplans are the same as floorplans, but an alternative definition of equivalence is used. Two mosaic floorplans are considered equivalent if and only if they can be converted to each other by sliding the line segments that divide the blocks.Mosaic floorplans are widely used in VLSI circuit design. An important problem in this area is to find short binary string representations of the set of n-block mosaic floorplans. The best known representation is the Quarter-State Sequence which uses 4n bits. This paper introduces a simple binary representation of n-block mosaic floorplan using 3n − 3 bits. It has been shown that any binary representation of n-block mosaic floorplans must use at least (3n − o(n)) bits. Therefore, the representation presented in this paper is optimal (up to an additive lower order term).Baxter permutations are a set of permutations defined by prohibited subsequences. Baxter permutations have been shown to have one-to-one correspondences to many interesting objects in the so-called Baxter combinatorial family. In particular, there exists a simple one-to-one correspondence between mosaic floorplans and Baxter permutations. As a result, the methods introduced in this paper also lead to an optimal binary representation of Baxter permutations and all objects in the Baxter combinatorial family.

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Algebraic and combinatorial structures on pairs of twin binary trees

Cited by 62 publications

References 27 publications

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Hopf algebra structure on packed square matrices

A simple optimal binary representation of mosaic floorplans and Baxter permutations

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