Rank, two-color partitions and Mock theta function

TL;DR

This paper connects partition counts with positive odd rank to two-color partitions with specific constraints, deriving new representations for mock theta functions and identities related to the smallest part partition function.

Contribution

It introduces novel combinatorial interpretations linking partitions and mock theta functions, including new representations and identities for these functions.

Findings

Equal counts of partitions with positive odd rank and specific two-color partitions.

New representation for the third order mock theta function f_3(q).

An analogue of the fundamental identity for Spt(n).

Abstract

In this paper, we establish that the number of partitions of a natural number with positive odd rank is equal to the number of two-color partitions (red and blue), where the smallest part is even (say $2n$) and all red parts are even and lie within the interval $(2n,4n]$. This led us to derive a new representation for the third order mock theta function $f_3(q)$ and an analogue of the fundamental identity for the smallest part partition function Spt$(n)$, both of which are of significant interest in their own right. We also consider the odd smallest part version of the above two-color partition, whose generating function involves another third order mock theta function $\phi_3(q)$.