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Abstract. We discuss a new companion to Capparelli's identities. Capparelli's identities for m = 1, 2 state that the number of partitions of n into distinct parts not congruent to m, −m modulo 6 is equal to the number of partitions of n into distinct parts not equal to m, where the difference between parts is greater than or equal to 4, unless consecutive parts are either both consecutive multiples of 3 or add up to to a multiple of 6. In this paper we show that the set of partitions of n into distinct parts where the odd-indexed parts are not congruent to m modulo 3, the even-indexed parts are not congruent to 3 − m modulo 3, and 3l + 1 and 3l + 2 do not appear together as consecutive parts for any integer l has the same number of elements as the above mentioned Capparelli's partitions of n. In this study we also extend the work of Alladi, Andrews and Gordon by providing a complete set of generating functions for the refined Capparelli partitions, and conjecture some combinatorial inequalities.

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-Szegő polynomials.

We study the generating functions for cylindric partitions with profile (c 1 , c 2 , c 3 ) for all c 1 , c 2 , c 3 such that c 1 + c 2 + c 3 = 5. This allows us to discover and prove seven new A 2 Rogers-Ramanujan identities modulo 8 with quadruple sums, related with work of Andrews, Schilling, and Warnaar.