47 research outputs found
Actions on permutations and unimodality of descent polynomials
We study a group action on permutations due to Foata and Strehl and use it to
prove that the descent generating polynomial of certain sets of permutations
has a nonnegative expansion in the basis ,
. This property implies symmetry and unimodality. We
prove that the action is invariant under stack-sorting which strengthens recent
unimodality results of B\'ona. We prove that the generalized permutation
patterns and are invariant under the action and use this to
prove unimodality properties for a -analog of the Eulerian numbers recently
studied by Corteel, Postnikov, Steingr\'{\i}msson and Williams.
We also extend the action to linear extensions of sign-graded posets to give
a new proof of the unimodality of the -Eulerian polynomials of
sign-graded posets and a combinatorial interpretations (in terms of
Stembridge's peak polynomials) of the corresponding coefficients when expanded
in the above basis.
Finally, we prove that the statistic defined as the number of vertices of
even height in the unordered decreasing tree of a permutation has the same
distribution as the number of descents on any set of permutations invariant
under the action. When restricted to the set of stack-sortable permutations we
recover a result of Kreweras.Comment: 19 pages, revised version to appear in Europ. J. Combi
Distributive lattice models of the type C one-rowed Weyl group symmetric functions
We present two families of diamond-colored distributive lattices â one known and one new â that we can show are models of the type C one-rowed Weyl symmetric functions. These lattices are constructed using certain sequences of positive integers that are visualized as ïŹlling the boxes of one-rowed partition diagrams. We show how natural orderings of these one-rowed tableaux produce our distributive lattices as sublattices of a more general object, and how a natural coloring of the edges of the associated order diagrams yields a certain diamond-coloring property. We show that each edge-colored lattice possesses a certain structure that is associated with the type C Weyl groups. Moreover, we produce a bijection that shows how any two aïŹliated lattices, one from each family, are models for the same type C one-rowed Weyl symmetric function. While our type C one-rowed lattices have multiple algebraic contexts, this thesis largely focusses on their combinatorial aspects
On positivity of Ehrhart polynomials
Ehrhart discovered that the function that counts the number of lattice points
in dilations of an integral polytope is a polynomial. We call the coefficients
of this polynomial Ehrhart coefficients, and say a polytope is Ehrhart positive
if all Ehrhart coefficients are positive (which is not true for all integral
polytopes). The main purpose of this article is to survey interesting families
of polytopes that are known to be Ehrhart positive and discuss the reasons from
which their Ehrhart positivity follows. We also include examples of polytopes
that have negative Ehrhart coefficients and polytopes that are conjectured to
be Ehrhart positive, as well as pose a few relevant questions.Comment: 40 pages, 7 figures. To appear in in Recent Trends in Algebraic
Combinatorics, a volume of the Association for Women in Mathematics Series,
Springer International Publishin
Three Fuss-Catalan posets in interaction and their associative algebras
We introduce -cliffs, a generalization of permutations and increasing
trees depending on a range map . We define a first lattice structure on
these objects and we establish general results about its subposets. Among them,
we describe sufficient conditions to have EL-shellable posets, lattices with
algorithms to compute the meet and the join of two elements, and lattices
constructible by interval doubling. Some of these subposets admit natural
geometric realizations. Then, we introduce three families of subposets which,
for some maps , have underlying sets enumerated by the Fuss-Catalan
numbers. Among these, one is a generalization of Stanley lattices and another
one is a generalization of Tamari lattices. These three families of posets fit
into a chain for the order extension relation and they share some properties.
Finally, in the same way as the product of the Malvenuto-Reutenauer algebra
forms intervals of the right weak order of permutations, we construct algebras
whose products form intervals of the lattices of -cliff. We provide
necessary and sufficient conditions on to have associative, finitely
presented, or free algebras. We end this work by using the previous
Fuss-Catalan posets to define quotients of our algebras of -cliffs. In
particular, one is a generalization of the Loday-Ronco algebra and we get new
generalizations of this structure.Comment: 63 page
Polynomial maps with applications to combinatorics and probability theory
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1994.Includes bibliographical references (leaves 79-80).by Dan Port.Ph.D