376 research outputs found
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
Partitions of Matrix Spaces With an Application to -Rook Polynomials
We study the row-space partition and the pivot partition on the matrix space
. We show that both these partitions are reflexive
and that the row-space partition is self-dual. Moreover, using various
combinatorial methods, we explicitly compute the Krawtchouk coefficients
associated with these partitions. This establishes MacWilliams-type identities
for the row-space and pivot enumerators of linear rank-metric codes. We then
generalize the Singleton-like bound for rank-metric codes, and introduce two
new concepts of code extremality. Both of them generalize the notion of MRD
codes and are preserved by trace-duality. Moreover, codes that are extremal
according to either notion satisfy strong rigidity properties analogous to
those of MRD codes. As an application of our results to combinatorics, we give
closed formulas for the -rook polynomials associated with Ferrers diagram
boards. Moreover, we exploit connections between matrices over finite fields
and rook placements to prove that the number of matrices of rank over
supported on a Ferrers diagram is a polynomial in , whose
degree is strictly increasing in . Finally, we investigate the natural
analogues of the MacWilliams Extension Theorem for the rank, the row-space, and
the pivot partitions
Symmetric Grothendieck polynomials, skew Cauchy identities, and dual filtered Young graphs
Symmetric Grothendieck polynomials are analogues of Schur polynomials in the
K-theory of Grassmannians. We build dual families of symmetric Grothendieck
polynomials using Schur operators. With this approach we prove skew Cauchy
identity and then derive various applications: skew Pieri rules, dual
filtrations of Young's lattice, generating series and enumerative identities.
We also give a new explanation of the finite expansion property for products of
Grothendieck polynomials
Pieri rules for the K-theory of cominuscule Grassmannians
We prove Pieri formulas for the multiplication with special Schubert classes
in the K-theory of all cominuscule Grassmannians. For Grassmannians of type A
this gives a new proof of a formula of Lenart. Our formula is new for
Lagrangian Grassmannians, and for orthogonal Grassmannians it proves a special
case of a conjectural Littlewood-Richardson rule of Thomas and Yong. Recent
work of Thomas and Yong and of E. Clifford has shown that the full
Littlewood-Richardson rule for orthogonal Grassmannians follows from the Pieri
case proved here. We describe the K-theoretic Pieri coefficients both as
integers determined by positive recursive identities and as the number of
certain tableaux. The proof is based on a computation of the sheaf Euler
characteristic of triple intersections of Schubert varieties, where at least
one Schubert variety is special
Stammering tableaux
The PASEP (Partially Asymmetric Simple Exclusion Process) is a probabilistic
model of moving particles, which is of great interest in combinatorics, since
it appeared that its partition function counts some tableaux. These tableaux
have several variants such as permutations tableaux, alternative tableaux,
tree- like tableaux, Dyck tableaux, etc. We introduce in this context certain
excursions in Young's lattice, that we call stammering tableaux (by analogy
with oscillating tableaux, vacillating tableaux, hesitating tableaux). Some
natural bijections make a link with rook placements in a double staircase,
chains of Dyck paths obtained by successive addition of ribbons, Laguerre
histories, Dyck tableaux, etc.Comment: Clarification and better exposition thanks reviewer's report
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