Hopf Algebra of the Planar Binary Trees

International audience Generalizing the connection between the classes of the sylvester congruence and the binary trees, we show that the classes of the congruence of the weak order on Sn defined as the transitive closure of the rewriting rule UacV1b1 ···VkbkW ≡k UcaV1b1 ···VkbkW, for letters a < b1,...,bk < c and words U,V1,...,Vk,W on [n], are in bijection with acyclic k-triangulations of the (n + 2k)-gon, or equivalently with acyclic pipe dreams for the permutation (1,...,k,n + k,...,k + 1,n + k + 1,...,n + 2k). It enables us to transport the known lattice and Hopf algebra structures from the congruence classes of ≡k to these acyclic pipe dreams, and to describe the product and coproduct of this algebra in terms of pipe dreams. Moreover, it shows that the fan obtained by coarsening the Coxeter fan according to the classes of ≡k is the normal fan of the corresponding brick polytope

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Topologies of locally minimal planar binary trees

Russian Mathematical Surveys ◽

10.1070/rm1994v049n06abeh002453 ◽

1994 ◽

Vol 49 (6) ◽

pp. 201-202

Author(s):

A O Ivanov ◽

A A Tuzhilin

Keyword(s):

Binary Trees ◽

Planar Binary Trees

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Lattice of combinatorial Hopf algebras: binary trees with multiplicities

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2372 ◽

2013 ◽

Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽

Author(s):

Jean-Baptiste Priez

Keyword(s):

Hopf Algebra ◽

Hopf Algebras ◽

Lattice Structure ◽

Binary Trees ◽

Hook Length ◽

Combinatorial Hopf Algebras ◽

Combinatorial Hopf Algebra ◽

International Audience ◽

Comme Application ◽

Premiere Partie

International audience In a first part, we formalize the construction of combinatorial Hopf algebras from plactic-like monoids using polynomial realizations. Thank to this construction we reveal a lattice structure on those combinatorial Hopf algebras. As an application, we construct a new combinatorial Hopf algebra on binary trees with multiplicities and use it to prove a hook length formula for those trees. Dans une première partie, nous formalisons la construction d’algèbres de Hopf combinatoires à partir d’une réalisation polynomiale et de monoïdes de type monoïde plaxique. Grâce à cette construction, nous mettons à jour une structure de treillis sur ces algèbres de Hopf combinatoires. Comme application, nous construisons une nouvelle algèbre de Hopf sur des arbres binaires à multiplicités et on l’utilise pour démontrer une formule des équerressur ces arbres.

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The Cambrian Hopf Algebra

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2533 ◽

2015 ◽

Vol DMTCS Proceedings, 27th... (Proceedings) ◽

Author(s):

G. Chatel ◽

V. Pilaud

Keyword(s):

Hopf Algebra ◽

Binary Search ◽

Binary Trees ◽

Search Trees ◽

Local Conditions ◽

Bijective Correspondence ◽

Combinatorial Properties ◽

Labeled Trees ◽

Signed Permutations ◽

International Audience

International audience Cambrian trees are oriented and labeled trees which fulfill local conditions around each node generalizing the conditions for classical binary search trees. Based on the bijective correspondence between signed permutations and leveled Cambrian trees, we define the Cambrian Hopf algebra generalizing J.-L. Loday and M. Ronco’s algebra on binary trees. We describe combinatorially the products and coproducts of both the Cambrian algebra and its dual in terms of operations on Cambrian trees. Finally, we define multiplicative bases of the Cambrian algebra and study structural and combinatorial properties of their indecomposable elements. Les arbres Cambriens sont des arbres orientés et étiquetés qui satisfont des conditions locales autour de leurs nœuds généralisant les conditions des arbres binaires de recherche classiques. A partir de la correspondence bijective entre permutations signées et arbres Cambriens à niveau, nous définissons l’algèbre Cambrienne qui généralise l’algèbre sur les arbres binaires de J.-L. Loday et M. Ronco. Nous donnons une description combinatoire du produit et du coproduit aussi bien dans l’algèbre Cambrienne que dans sa duale en termes d’opérations sur les arbres Cambriens. Enfin, nous définissons des bases multiplicatives de l’algèbre Cambrienne et étudions les propriétés structurelles et énumératives de leurs éléments indécomposables.

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