7,131 research outputs found
Superization and (q,t)-specialization in combinatorial Hopf algebras
We extend a classical construction on symmetric functions, the superization
process, to several combinatorial Hopf algebras, and obtain analogs of the
hook-content formula for the (q,t)-specializations of various bases. Exploiting
the dendriform structures yields in particular (q,t)-analogs of the
Bjorner-Wachs q-hook-length formulas for binary trees, and similar formulas for
plane trees.Comment: 30 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
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
Brick polytopes, lattice quotients, and Hopf algebras
This paper is motivated by the interplay between the Tamari lattice, J.-L.
Loday's realization of the associahedron, and J.-L. Loday and M. Ronco's Hopf
algebra on binary trees. We show that these constructions extend in the world
of acyclic -triangulations, which were already considered as the vertices of
V. Pilaud and F. Santos' brick polytopes. We describe combinatorially a natural
surjection from the permutations to the acyclic -triangulations. We show
that the fibers of this surjection are the classes of the congruence
on defined as the transitive closure of the rewriting rule for letters
and words on . We then
show that the increasing flip order on -triangulations is the lattice
quotient of the weak order by this congruence. Moreover, we use this surjection
to define a Hopf subalgebra of C. Malvenuto and C. Reutenauer's Hopf algebra on
permutations, indexed by acyclic -triangulations, and to describe the
product and coproduct in this algebra and its dual in term of combinatorial
operations on acyclic -triangulations. Finally, we extend our results in
three directions, describing a Cambrian, a tuple, and a Schr\"oder version of
these constructions.Comment: 59 pages, 32 figure
The # product in combinatorial Hopf algebras
We show that the # product of binary trees introduced by Aval and Viennot
[arXiv:0912.0798] is in fact defined at the level of the free associative
algebra, and can be extended to most of the classical combinatorial Hopf
algebras.Comment: 20 page
Trees, functional equations, and combinatorial Hopf algebras
One of the main virtues of trees is to represent formal solutions of various
functional equations which can be cast in the form of fixed point problems.
Basic examples include differential equations and functional (Lagrange)
inversion in power series rings. When analyzed in terms of combinatorial Hopf
algebras, the simplest examples yield interesting algebraic identities or
enumerative results.Comment: 14 pages, LaTE
Structure of the Malvenuto-Reutenauer Hopf algebra of permutations (Extended Abstract)
We analyze the structure of the Malvenuto-Reutenauer Hopf algebra of
permutations in detail. We give explicit formulas for its antipode, prove that
it is a cofree coalgebra, determine its primitive elements and its coradical
filtration and show that it decomposes as a crossed product over the Hopf
algebra of quasi-symmetric functions. We also describe the structure constants
of the multiplication as a certain number of facets of the permutahedron. Our
results reveal a close relationship between the structure of this Hopf algebra
and the weak order on the symmetric groups.Comment: 12 pages, 2 .eps figures. (minor revisions) Extended abstract for
Formal Power Series and Algebraic Combinatorics, Melbourne, July 200
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