The Andrews-gordon Identities and Q-multinomial Coefficients

We prove polynomial boson-fermion identities for the generating function of the number of partitions of n of the form n = ∑L−1 j=1 jfj, with f1 ≤ i−1, fL−1 ≤ i ′−1 and fj+fj+1 ≤ k. The bosonic side of the identities involves q-deformations of the coefficients of xa in the expansion of (1 + x + · · · + xk)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 → ∞, our identities reproduce the analytic form of Gordon’s generalization of the Rogers–Ramanujan identities, as found by Andrews. Using the q → 1/q duality, identities are obtained for branching functions corresponding to cosets of type (A (1) 1 )k×(A (1) 1 )l/(A (1) 1 )k+l of fractional level l. ∗e-mail: [email protected]