Divisibility Properties of Odd Part and Even Part Sums in Cubic Partitions

Introduction

Throughout this paper, for complex numbers $a$, $b$, and $q$ with $|q| < 1$, we use the standard notation

For convenience, we write $f_r=(q^r;q^r)_\infty$, where $r$ is a positive integer.

A non-increasing finite sequence of positive integers $\beta_1 \ge\beta_2 \ge\cdots\ge\beta_s > 0$ is said to be a partition of a positive integer $n$ if $n=\sum_{i=1}^{s}\beta_i$. The integers $\beta_i$ are called the parts of the partition. If $p(n)$ denotes the number of partitions of $n$, then its generating function [7] is given by

For example, $p(7)=15$, and the corresponding partitions are $7$, $6+1$, $5+2$, $5+1+1$, $4+3$, $4+2+1$, $4+1+1+1$, $3+3+1$, $3+2+2$, $3+2+1+1$, $3+1+1+1+1$, $2+2+2+1$, $2+2+1+1+1$, $2+1+1+1+1+1$, and $1+1+1+1+1+1+1$. Among the most celebrated arithmetic properties of $p(n)$ are the following congruences due to Ramanujan [11]:

valid for every nonnegative integer $n$. Ramanujan’s congruences have motivated extensive investigations of divisibility properties of several restricted partition functions.

Chan [5] studied the cubic partition function $a(n)$, in which even parts may appear in two distinct colors. The generating function of $a(n)$ is given by

For example, the cubic partitions of 3 are $3$, $2_r+1$, $2_g+1$, and $1+1+1$, where the subscripts $r$ and $g$ distinguish the two colors of the even part. Clearly, $a(3)=4$. Chan [5 Theorem 1] proved the identity

which immediately gives the Ramanujan-type congruence $a(3n+2)\equiv0\pmod{3}$. Chen and Lin [6 Theorems 1.3–1.4] proved that, for every $n\geq0$,

and

Xiong [14 Theorem 1.3] extended the modulo 5 result to an infinite family modulo powers of 5.

Recently, Andrews and Ghosh Dastidar [1] defined $SOME(n)$ as the difference between the sum of all odd parts and the sum of all even parts in the partitions of a nonnegative integer $n$. They established several divisibility properties of $SOME(n)$. In particular, they proved [1 Theorem 4] that $SOME(5n+4) \equiv0 \pmod{5}$, which, together with Ramanujan’s congruence $p(5n+4) \equiv0 \pmod{5}$, implies that both the sum of all odd parts and the sum of all even parts in the partitions of $5n+4$ are divisible by 5. They also conjectured that $SOME(\lambda) \equiv0 \pmod{5^\alpha}$, where $\alpha$ is a positive integer and $\lambda$ is a nonnegative integer satisfying $24\lambda\equiv1 \pmod{5^\alpha}$, which was subsequently proved by Bardhan and Saikia [2]. Furthermore, Garvan and Sarma [8] studied sums of certain parts in overpartitions and partitions without repeated odd parts. Subsequently, Baruah and Gogoi [3] investigated analogous sums of overlined parts in overpartitions.

Motivated by the $SOME(n)$ function and its properties, in this paper, we define the corresponding arithmetic functions $COS(n)$ and $CES(n)$ for cubic partitions and study their divisibility properties. The function $COS(n)$ (resp. $CES(n)$) denotes the sum of all odd (resp. even) parts in all cubic partitions of a positive integer $n$. Since the sum of all parts of each cubic partition of $n$ is $n$, it follows that

For example, $COS(3)=8$ and $CES(3)=4$. For convenience, we set $COS(0)=CES(0)=0$.

The paper is organized as follows. § 2 records the necessary $q$-identities and extraction formulas used in the subsequent proofs. In § 3, we derive the generating functions and divisor-sum formulas for $COS(n)$ and $CES(n)$ and establish their congruences modulo 3, 4, 5, and 7. Finally, § 4 presents a conjecture for $COS(n)$ and $CES(n)$ modulo higher prime powers.

Preliminary results

Ramanujan’s general theta function is defined by [4 Eq. (1.2.1), p. 6]

Three important special cases [4 Eqs. (1.2.2)–(1.2.4), pp. 6–7] are

and

The product representations in (2.1), (2.2), and (2.3) follow from Jacobi’s triple product identity [4 Theorem 1.3.3], given by

which can be equivalently stated as

Replacing $q$ by $-q$ in (2.1) and simplifying, we obtain

Replacing $q$ by $q^2$ in (2.2), we obtain