137 research outputs found
Ideals of Quasi-Symmetric Functions and Super-Covariant Polynomials for S_n
The aim of this work is to study the quotient ring R_n of the ring
Q[x_1,...,x_n] over the ideal J_n generated by non-constant homogeneous
quasi-symmetric functions. We prove here that the dimension of R_n is given by
C_n, the n-th Catalan number. This is also the dimension of the space SH_n of
super-covariant polynomials, that is defined as the orthogonal complement of
J_n with respect to a given scalar product. We construct a basis for R_n whose
elements are naturally indexed by Dyck paths. This allows us to understand the
Hilbert series of SH_n in terms of number of Dyck paths with a given number of
factors.Comment: LaTeX, 3 figures, 12 page
On the diagram of 132-avoiding permutations
The diagram of a 132-avoiding permutation can easily be characterized: it is
simply the diagram of a partition. Based on this fact, we present a new
bijection between 132-avoiding and 321-avoiding permutations. We will show that
this bijection translates the correspondences between these permutations and
Dyck paths given by Krattenthaler and by Billey-Jockusch-Stanley, respectively,
to each other. Moreover, the diagram approach yields simple proofs for some
enumerative results concerning forbidden patterns in 132-avoiding permutations.Comment: 20 pages; additional reference is adde
Ad-nilpotent ideals of a Borel subalgebra: generators and duality
It was shown by Cellini and Papi that an ad-nilpotent ideal determines
certain element of the affine Weyl group, and that there is a bijection between
the ad-nilpotent ideals and the integral points of a simplex with rational
vertices. We give a description of the generators of ad-nilpotent ideals in
terms of these elements, and show that an ideal has generators if and only
it lies on the face of this simplex of codimension . We also consider two
combinatorial statistics on the set of ad-nilpotent ideals: the number of
simple roots in the ideal and the number of generators. Considering the first
statistic reveals some relations with the theory of clusters
(Fomin-Zelevinsky). The distribution of the second statistic suggests that
there should exist a natural involution (duality) on the set of ad-nilpotent
ideals. Such an involution is constructed for the series A,B,C.Comment: LaTeX2e, 23 page
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Enumerative Combinatorics
Enumerative Combinatorics focusses on the exact and asymptotic counting of combinatorial objects. It is strongly connected to the probabilistic analysis of large combinatorial structures and has fruitful connections to several disciplines, including statistical physics, algebraic combinatorics, graph theory and computer science. This workshop brought together experts from all these various fields, including also computer algebra, with the goal of promoting cooperation and interaction among researchers with largely varying backgrounds
A combinatorial model for computing volumes of flow polytopes
We introduce new families of combinatorial objects whose enumeration computes
volumes of flow polytopes. These objects provide an interpretation, based on
parking functions, of Baldoni and Vergne's generalization of a volume formula
originally due to Lidskii. We recover known flow polytope volume formulas and
prove new volume formulas for flow polytopes that were seemingly
unapproachable. A highlight of our model is an elegant formula for the flow
polytope of a graph we call the caracol graph.
As by-products of our work, we uncover a new triangle of numbers that
interpolates between Catalan numbers and the number of parking functions, we
prove the log-concavity of rows of this triangle along with other sequences
derived from volume computations, and we introduce a new Ehrhart-like
polynomial for flow polytope volume and conjecture product formulas for the
polytopes we consider.Comment: 34 pages, 15 figures. v2: updated after referee reports; includes a
proof of Proposition 8.7. Accepted into Transactions of the AM
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