12,020 research outputs found
The Andrews-Gordon identities and -multinomial coefficients
We prove polynomial boson-fermion identities for the generating function of
the number of partitions of of the form , with
, and . The bosonic side of
the identities involves -deformations of the coefficients of in the
expansion of . A combinatorial interpretation for these
-multinomial coefficients is given using Durfee dissection partitions. The
fermionic side of the polynomial identities arises as the partition function of
a one-dimensional lattice-gas of fermionic particles. In the limit
, our identities reproduce the analytic form of Gordon's
generalization of the Rogers--Ramanujan identities, as found by Andrews. Using
the duality, identities are obtained for branching functions
corresponding to cosets of type of fractional level .Comment: 31 pages, Latex, 9 Postscript figure
Supernomial coefficients, polynomial identities and -series
-Analogues of the coefficients of in the expansion of are proposed. Useful properties, such as recursion
relations, symmetries and limiting theorems of the ``-supernomial
coefficients'' are derived, and a combinatorial interpretation using
generalized Durfee dissection partitions is given. Polynomial identities of
boson-fermion-type, based on the continued fraction expansion of and
involving the -supernomial coefficients, are proven. These include
polynomial analogues of the Andrews-Gordon identities. Our identities unify and
extend many of the known boson-fermion identities for one-dimensional
configuration sums of solvable lattice models, by introducing multiple
finitization parameters.Comment: 34 pages, Latex2e, figures; improved versio
On double sum generating functions in connection with some classical partition theorems
We focus on writing closed forms of generating functions for the number of
partitions with gap conditions as double sums starting from a combinatorial
construction. Some examples of the sets of partitions with gap conditions to be
discussed here are the set of Rogers--Ramanujan, G\"ollnitz--Gordon, and little
G\"ollnitz partitions. This work also includes finding the finite analogs of
the related generating functions and the discussion of some related series and
polynomial identities. Additionally, we present a different construction and a
double sum representation for the products similar to the ones that appear in
the Rogers--Ramanujan identities.Comment: 20 page
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
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