A combinatorial proof of the Rogers-Ramanujan and Schur identities

We give a combinatorial proof of the first Rogers-Ramanujan identity by using two symmetries of a new generalization of Dyson's rank. These symmetries are established by direct bijections.Comment: 12 pages, 5 figures; incorporated referee suggestions, simplified definition of (k,m)-rank, to appear in JCT(A

Bilateral identities of the Rogers-Ramanujan type

We derive by analytic means a number of bilateral identities of the Rogers-Ramanujan type. Our results include bilateral extensions of the Rogers-Ramanujan and the G\"ollnitz-Gordon identities, and of related identities by Ramanujan, Jackson, and Slater. We give corresponding results for multiseries including multilateral extensions of the Andrews-Gordon identities, of Bressoud's even modulus identities, and other identities. The here revealed closed form bilateral and multilateral summations appear to be the very first of their kind. Given that the classical Rogers-Ramanujan identities have well-established connections to various areas in mathematics and in physics, it is natural to expect that the new bilateral and multilateral identities can be similarly connected to those areas. This is supported by concrete combinatorial interpretations for a collection of four bilateral companions to the classical Rogers-Ramanujan identities.Comment: 25 page

Hall-Littlewood polynomials and characters of affine Lie algebras

The Weyl-Kac character formula gives a beautiful closed-form expression for the characters of integrable highest-weight modules of Kac-Moody algebras. It is not, however, a formula that is combinatorial in nature, obscuring positivity. In this paper we show that the theory of Hall-Littlewood polynomials may be employed to prove Littlewood-type combinatorial formulas for the characters of certain highest weight modules of the affine Lie algebras C_n^{(1)}, A_{2n}^{(2)} and D_{n+1}^{(2)}. Through specialisation this yields generalisations for B_n^{(1)}, C_n^{(1)}, A_{2n-1}^{(2)}, A_{2n}^{(2)} and D_{n+1}^{(2)} of Macdonald's identities for powers of the Dedekind eta-function. These generalised eta-function identities include the Rogers-Ramanujan, Andrews-Gordon and G\"ollnitz-Gordon q-series as special, low-rank cases.Comment: 33 pages, proofs of several conjectures from the earlier version have been include

The Andrews-Gordon identities and qq-multinomial coefficients

We prove polynomial boson-fermion identities for the generating function of the number of partitions of $n$ of the form $n=\sum_{j=1}^{L-1} j f_j$, with $f_1\leq i-1$, $f_{L-1} \leq i'-1$ and $f_j+f_{j+1}\leq k$. The bosonic side of the identities involves $q$-deformations of the coefficients of $x^a$ in the expansion of $(1+x+\cdots+ x^k)^L$. A combinatorial interpretation for these $q$-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 $L\to\infty$, our identities reproduce the analytic form of Gordon's generalization of the Rogers--Ramanujan identities, as found by Andrews. Using the $q \to 1/q$ duality, identities are obtained for branching functions corresponding to cosets of type $({\rm A}^{(1)}_1)_k \times ({\rm A}^{(1)}_1)_{\ell} / ({\rm A}^{(1)}_1)_{k+\ell}$ of fractional level $\ell$.Comment: 31 pages, Latex, 9 Postscript figure

The Method of Combinatorial Telescoping

We present a method for proving q-series identities by combinatorial telescoping, in the sense that one can transform a bijection or a classification of combinatorial objects into a telescoping relation. We shall illustrate this method by giving a combinatorial proof of Watson's identity which implies the Rogers-Ramanujan identities.Comment: 11 pages, 5 figures; to appear in J. Combin. Theory Ser.