Hopf Algebras of m-permutations, (m+1)-ary trees, and m-parking functions

The m-Tamari lattice of F. Bergeron is an analogue of the clasical Tamari order defined on objects counted by Fuss-Catalan numbers, such as m-Dyck paths or (m+1)-ary trees. On another hand, the Tamari order is related to the product in the Loday-Ronco Hopf algebra of planar binary trees. We introduce new combinatorial Hopf algebras based on (m+1)-ary trees, whose structure is described by the m-Tamari lattices. In the same way as planar binary trees can be interpreted as sylvester classes of permutations, we obtain (m+1)-ary trees as sylvester classes of what we call m-permutations. These objects are no longer in bijection with decreasing (m+1)-ary trees, and a finer congruence, called metasylvester, allows us to build Hopf algebras based on these decreasing trees. At the opposite, a coarser congruence, called hyposylvester, leads to Hopf algebras of graded dimensions (m+1)^{n-1}, generalizing noncommutative symmetric functions and quasi-symmetric functions in a natural way. Finally, the algebras of packed words and parking functions also admit such m-analogues, and we present their subalgebras and quotients induced by the various congruences.Comment: 51 page

Commutative combinatorial Hopf algebras

We propose several constructions of commutative or cocommutative Hopf algebras based on various combinatorial structures, and investigate the relations between them. A commutative Hopf algebra of permutations is obtained by a general construction based on graphs, and its non-commutative dual is realized in three different ways, in particular as the Grossman-Larson algebra of heap ordered trees. Extensions to endofunctions, parking functions, set compositions, set partitions, planar binary trees and rooted forests are discussed. Finally, we introduce one-parameter families interpolating between different structures constructed on the same combinatorial objects.Comment: 29 pages, LaTEX; expanded and updated version of math.CO/050245

A Hopf algebra of parking functions

If the moments of a probability measure on $\R$ are interpreted as a specialization of complete homogeneous symmetric functions, its free cumulants are, up to sign, the corresponding specializations of a sequence of Schur positive symmetric functions $(f_n)$. We prove that $(f_n)$ is the Frobenius characteristic of the natural permutation representation of \SG_n on the set of prime parking functions. This observation leads us to the construction of a Hopf algebra of parking functions, which we study in some detail.Comment: AmsLatex, 14 page

The Algebra of Binary Search Trees

We introduce a monoid structure on the set of binary search trees, by a process very similar to the construction of the plactic monoid, the Robinson-Schensted insertion being replaced by the binary search tree insertion. This leads to a new construction of the algebra of Planar Binary Trees of Loday-Ronco, defining it in the same way as Non-Commutative Symmetric Functions and Free Symmetric Functions. We briefly explain how the main known properties of the Loday-Ronco algebra can be described and proved with this combinatorial point of view, and then discuss it from a representation theoretical point of view, which in turns leads to new combinatorial properties of binary trees.Comment: 49 page

Expressão comparativa do gene Gus com dois diferentes promotores em citros transgênicos.

O objetivo deste trabalho foi comparar a expressão do gene repórter uidA(GUS), realizando-se para isso a transformação genética, via Agrobacterium tumefaciens, de plantas de citros (citrange ?Carrizo?) utilizando duas construções gênicas, sendo uma delas com promotor constitutivo, amplamente empregado em transgenia, e outra com promotor floema-específico, isolado e caracterizado pelo nosso grupo, para confirmar a especificidade deste promotor. Segmentos de epicótilo de plântulas germinadas in vitro foram utilizados como explantes. Os explantes foram inoculados com o inóculo contendo a bactéria com a construção gênica. Os explantes foram co-cultivados em meio contendo acetoseringona por 3 dias a 24°C, no escuro, e posteriormente transferidos para meio MS suplementado com canamicina e mantidos no escuro por aproximadamente 2 semanas a 28°C. Após esse período, as placas com os explantes foram colocadas sob a luz, e o meio foi trocado a cada 3-4 semanas, até os explantes começarem a formar brotos adequados para realizar o teste histoquímico com X-GLUC. A transformação dos explantes foi confirmada através do teste histoquímico e as taxas de eficiência de transformação encontradas foram 2,3% para o promotor floema-específico e 3,1% para o promotor constitutivo. A expressão do gene uidA utilizando a construção com o promotor floema-específico evidenciou preferência ao floema, confirmando a hipótese de que o promotor isolado era específico para essa região.Anais Web