Identities involving partitions with distinct odd parts and no parts congruent to 2 mod 4

TL;DR

The paper investigates identities linking partitions into distinct odd parts with partitions whose odd parts are distinct and with partitions having no parts congruent to $2\bmod 4$. It develops $q$-series proofs for three main identities and complements them with combinatorial bijections to establish the same equalities analytically and combinatorially. It also derives a generating-function identity $\sum_{n\ge0} c(n) q^n = \sum_{n\ge0} pod(n) q^n = \frac{(-q;q^2)_\infty}{(q^2;q^2)_\infty}$ and related forms, and extends the theory with explicit constructions and examples. The results provide a dual perspective on restricted partitions, connecting analytic $q$-series techniques with explicit combinatorial bijections, and contribute tools for studying partitions with constrained parity and multiplicity.

Abstract

Recently, Pankaj Jyoti Mahanta and Manjil P. Saikika proved some identities relating certain restricted partitions into distinct odd parts with the partition whose odd parts are distinct combinatorially. They asked for the q-series proofs. In this paper, we give q-series proofs of these identities. Also, the number of partitions into distinct odd parts equals to the number of the partitions with no parts congruent to 2 mod 4, so we can get some identities. We also give combinatorial proofs of these identities.