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 $\ell$.
Categories: q-alg hep-th math.QA