  This article is an extensive study of partitions with fixed number of odd and
even-indexed odd parts. We use these partitions to generalize recent results of
C. Savage and A. Sills. Moreover, we derive explicit formulas for generating
functions for partitions with bounds on the largest part, the number of parts
and with a fixed value of BG-rank or with a fixed value of alternating sum of
parts. We extend the work of C. Boulet, and as a result, obtain a four-variable
generalization of Gaussian binomial coefficients. In addition we provide
combinatorial interpretation of the Berkovich-Warnaar identity for Rogers-Szego
polynomials.
