On the Number of Partitions with Designated Summands

TL;DR

This paper explores the properties of the partition function PD(n) with designated summands, deriving new identities and formulas, and providing combinatorial interpretations of key congruences using advanced theta function techniques.

Contribution

It introduces explicit formulas for the generating functions of PD(3n) and PD(3n+1), and offers a combinatorial interpretation of a known divisibility congruence.

Findings

Derived a Ramanujan-type identity for PD(3n+2)

Found explicit formulas for generating functions of PD(3n) and PD(3n+1)

Provided a combinatorial interpretation of the divisibility congruence

Abstract

Andrews, Lewis and Lovejoy introduced the partition function PD(n) as the number of partitions of $n$ with designated summands, where we assume that among parts with equal size, exactly one is designated. They proved that PD(3n+2) is divisible by 3. We obtain a Ramanujan type identity for the generating function of PD(3n+2) which implies the congruence of Andrews, Lewis and Lovejoy. For PD(3n), Andrews, Lewis and Lovejoy showed that the generating function can be expressed as an infinite product of powers of $(1-q^{2n+1})$ times a function $F(q^2)$. We find an explicit formula for $F(q^2)$, which leads to a formula for the generating function of PD(3n). We also obtain a formula for the generating function of PD(3n+1). Our proofs rely on Chan's identity on Ramanujan's cubic continued fraction and some identities on cubic theta functions. By introducing a rank for the partitions with designed summands, we give a combinatorial interpretation of the congruence of Andrews, Lewis and Lovejoy.