12,020 research outputs found

    The Andrews-Gordon identities and qq-multinomial coefficients

    Full text link
    We prove polynomial boson-fermion identities for the generating function of the number of partitions of nn of the form n=j=1L1jfjn=\sum_{j=1}^{L-1} j f_j, with f1i1f_1\leq i-1, fL1i1f_{L-1} \leq i'-1 and fj+fj+1kf_j+f_{j+1}\leq k. The bosonic side of the identities involves qq-deformations of the coefficients of xax^a in the expansion of (1+x++xk)L(1+x+\cdots+ x^k)^L. A combinatorial interpretation for these qq-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 LL\to\infty, our identities reproduce the analytic form of Gordon's generalization of the Rogers--Ramanujan identities, as found by Andrews. Using the q1/qq \to 1/q duality, identities are obtained for branching functions corresponding to cosets of type (A1(1))k×(A1(1))/(A1(1))k+({\rm A}^{(1)}_1)_k \times ({\rm A}^{(1)}_1)_{\ell} / ({\rm A}^{(1)}_1)_{k+\ell} of fractional level \ell.Comment: 31 pages, Latex, 9 Postscript figure

    Supernomial coefficients, polynomial identities and qq-series

    Full text link
    qq-Analogues of the coefficients of xax^a in the expansion of j=1N(1+x+...+xj)Lj\prod_{j=1}^N (1+x+...+x^j)^{L_j} are proposed. Useful properties, such as recursion relations, symmetries and limiting theorems of the ``qq-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 p/kp/k and involving the qq-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

    Get PDF
    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 qq-Rook Polynomials

    Full text link
    We study the row-space partition and the pivot partition on the matrix space Fqn×m\mathbb{F}_q^{n \times m}. 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 qq-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 rr over Fq\mathbb{F}_q supported on a Ferrers diagram is a polynomial in qq, whose degree is strictly increasing in rr. Finally, we investigate the natural analogues of the MacWilliams Extension Theorem for the rank, the row-space, and the pivot partitions
    corecore