Sieved partition functions and $q$-binomial coefficients

The q-binomial coefficient is a polynomial in q. Given an integer t and a residue class r modulo t, a sieved q-binomial coefficient is the sum of those terms whose exponents are congruent to r modulo t. In this paper explicit polynomial identities in ${q^t}$ are given for sieved q-binomial coefficients. As a limiting case, generating functions for the sieved partition function are found as multidimensional theta functions. A striking corollary of this representation is the proof of Ramanujan’s congruences $\bmod 5, 7$, and 11 by exhibiting symmetry groups of orders 5, 7, and 11 of explicit quadratic forms. We also verify the Subbarao conjecture for $t = 3$, $t = 5$, and $t = 10$.