1,479 research outputs found
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 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 . We prove that 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
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