On partitions with fixed number of even-indexed and odd-indexed odd parts

TL;DR

This paper investigates partitions with fixed counts of odd and even-indexed odd parts, generalizing recent results, deriving explicit generating functions, and extending classical identities with combinatorial interpretations.

Contribution

It introduces new formulas for partitions with bounds and fixed parameters, generalizes Gaussian binomial coefficients, and interprets key identities combinatorially.

Findings

Derived explicit generating functions for constrained partitions

Extended Gaussian binomial coefficients to four variables

Provided combinatorial interpretation of Rogers-Szego polynomial identities

Abstract

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.