461 research outputs found
Combinatorics of r-Dyck paths, r-Parking functions, and the r-Tamari lattices
This paper's aim is to present recent combinatorial considerations on r-Dyck
paths, r-Parking functions, and the r-Tamari lattices. Giving a better
understanding of the combinatorics of these objects has become important in
view of their (conjectural) role in the description of the graded character of
the Sn-modules of bivariate and trivariate diagonal coinvariant spaces for the
symmetric group.Comment: 36 pages, 12 figure
The Matrix Ansatz, Orthogonal Polynomials, and Permutations
In this paper we outline a Matrix Ansatz approach to some problems of
combinatorial enumeration. The idea is that many interesting quantities can be
expressed in terms of products of matrices, where the matrices obey certain
relations. We illustrate this approach with applications to moments of
orthogonal polynomials, permutations, signed permutations, and tableaux.Comment: to appear in Advances in Applied Mathematics, special issue for
Dennis Stanto
A generating algorithm for ribbon tableaux and spin polynomials
We describe a general algorithm for generating various families of ribbon
tableaux and computing their spin polynomials. This algorithm is derived from a
new matricial coding. An advantage of this new notation lies in the fact that
it permits one to generate ribbon tableaux with skew shapes. This algorithm
permits us to compute quickly big LLT polynomials in MuPAD-Combinat
A Combinatorial Formula for Macdonald Polynomials
We prove a combinatorial formula for the Macdonald polynomial H_mu(x;q,t)
which had been conjectured by the first author. Corollaries to our main theorem
include the expansion of H_mu(x;q,t) in terms of LLT polynomials, a new proof
of the charge formula of Lascoux and Schutzenberger for Hall-Littlewood
polynomials, a new proof of Knop and Sahi's combinatorial formula for Jack
polynomials as well as a lifting of their formula to integral form Macdonald
polynomials, and a new combinatorial rule for the Kostka-Macdonald coefficients
K_{lambda,mu}(q,t) in the case that mu is a partition with parts less than or
equal to 2.Comment: 29 page
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