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.
- Pub Date:
- October 2015
- Mathematics - Number Theory;
- Mathematics - Combinatorics;
- 17 pages, 8 tables. The paper will appear in the Journal of Number Theory