---
title: Partition-Theoretic SOME Functions
url: https://www.emergentmind.com/topics/some
type: topic
---

# Partition-Theoretic SOME Functions

Searching arXiv for papers specifically about the partition-theoretic \(SOME(n)\), \(DSOME(n)\), and \(\overline{SOME}(n)\) functions.
I’m querying arXiv for exact matches and related partition-function terminology.
In partition theory, \(SOME(n)\) denotes the sum of all odd parts minus the sum of all even parts, taken over all ordinary partitions of a positive integer \(n\). Andrews and Dastidar introduced the functions \(SOME(n)\) and \(DSOME(n)\), with \(DSOME(n)\) defined analogously for distinct partitions, and Gireesh and Hemanthkumar introduced the overpartition analogue \(\overline{SOME}(n)\). Subsequent work established new identities, congruences, and monotonicity results for these functions, and also introduced two broader generalizations: a partition-family statistic \(S_{\mathcal P}(n)\) and a colour partition analogue [2606.31264].

## 1. Definitions and generating series

The basic statistics are defined as follows. \(SOME(n)\) is the sum of all odd parts minus the sum of all even parts over all ordinary partitions of \(n\). \(DSOME(n)\) is the same quantity for distinct partitions, and \(\overline{SOME}(n)\) is the corresponding quantity for overpartitions [2606.31264].

For \(SOME(n)\), the generating function previously established is
\[
\sum_{n=0}^\infty SOME(n) q^n = \frac{1}{(q;q)_\infty} \sum_{m=1}^\infty \frac{q^m}{(1+q^m)^2}.
\]

For \(DSOME(n)\), the generating function is
\[
\sum_{n=0}^\infty DSOME(n) q^n = (-q;q)_\infty \sum_{m=1}^\infty \frac{(-1)^{m-1} q^m}{(1+q^m)^2}.
\]

A closed form due to Baruah and Gogoi is also recorded:
\[
\sum_{n=0}^{\infty} DSOME(n) q^n = \frac{1}{8} \left( \frac{g_2}{g_1} - \frac{g_1^7}{g_2^3} \right),
\]
where \(g_t := (q^t;q^t)_\infty\).

A generating function for \(\overline{SOME}(n)\) was also previously established, and the 2026 study treats it alongside \(SOME(n)\) and \(DSOME(n)\) as part of a common framework of partition statistics [2606.31264].

## 2. Structural identity for \(SOME(n)\)

A central identity links \(SOME(n)\) to the partition function \(p(n)\) and the divisor-sum function \(\sigma(i)\):
\[
SOME(n) + 4 \sum_{i=1}^{\lfloor n/2 \rfloor} p(n - 2i) \sigma(i) = n p(n).
\]
Here \(p(n)\) is the partition function and \(\sigma(i)\) is the sum of divisors of \(i\) [2606.31264].

This identity places \(SOME(n)\) in direct relation with classical partition-theoretic and multiplicative arithmetic data. A plausible implication is that the statistic is not merely combinatorial bookkeeping over odd and even parts, but a quantity constrained by a precise convolutional structure involving partitions and divisor sums.

The same work records the previously known congruence \(SOME(4n) \equiv 0 \pmod{4}\), and later generalizes that pattern to arbitrary partition families through \(S_{\mathcal P}(n)\). In that sense, the identity above is both a specific formula for \(SOME(n)\) and a prototype for a broader divisibility theory [2606.31264].

## 3. \(DSOME(n)\), \(\overline{SOME}(n)\), and interrelations

For \(DSOME(n)\), an explicit divisor-sum formula is given:
\[
DSOME(n) = \sum_{\substack{1 \leq j \leq n\\ j \text{ odd}}} p_d(n-j) \sigma(j) - 3 \sum_{\substack{1 \leq j \leq n\\ j \text{ even}}} p_d(n-j) \sigma\left( \frac{j}{2^{\nu_2(j)}} \right),
\]
where \(p_d(n)\) counts distinct partitions and \(\nu_2(j)\) is the exponent of \(2\) in \(j\) [2606.31264].

A second identity expresses \(DSOME(n)\) in terms of \(SOME(n)\):
\[
DSOME(n) = \sum_{\substack{j \in \mathbb{Z}\\ j(3j-1) \leq n}} (-1)^j SOME(n - j(3j-1)) - 2 \sum_{\substack{j \geq 0,\ j(j+1)/2 \leq n,\, n \equiv j(j+1)/2 \pmod{2}}} SOME\left( \frac{n-j(j+1)/2}{2} \right).
\]

The three statistics are further connected by the relation
\[
\overline{SOME}(n) = SOME(n) + DSOME(n) + \sum_{\substack{r \geq 1,\, r(3r-1)/2 \leq n}} (-1)^{r+1} \overline{SOME}(n - r(3r-1)/2) + \sum_{\substack{r \geq 1,\, r(3r-1) \leq n}} (-1)^r SOME(n - r(3r-1)).
\]

These formulas show that the ordinary, distinct, and overpartition versions are not isolated statistics. They are tied together by identities involving pentagonal-type quadratic expressions. This suggests a common \(q\)-series and partition-theoretic substrate underlying the three functions.

## 4. Congruences and monotonicity

The 2026 results establish new congruences for \(\overline{SOME}(n)\). For any \(n \geq 1\),
\[
\overline{SOME}(n) \equiv
\begin{cases}
2 \pmod{4}, & \text{if } n \text{ is an odd perfect square},\\
0 \pmod{4}, & \text{otherwise},
\end{cases}
\]
and
\[
\overline{SOME}(n) \equiv
\begin{cases}
2 \pmod{8}, & \text{if } n \text{ is an odd perfect square},\\
0 \pmod{8}, & \text{otherwise}.
\end{cases}
\]
The same paper also gives, for any integer \(\alpha \ge 0\), an if and only if criterion characterizing when
\[
\overline{SOME}(n) \equiv 0 \pmod{2^{\alpha+2}}
\]
in terms of representation numbers \(r_{2^{\alpha+1}-2}(m)\) [2606.31264].

Monotonicity is established for both the ordinary and overpartition versions:
\[
SOME(n) \geq SOME(n-2)\quad \text{for all } n \geq 2,
\]
and
\[
\overline{SOME}(n) \geq \overline{SOME}(n-2)\qquad \forall n \geq 2.
\]

Two consequences are stated explicitly for \(SOME(n)\): both sequences \(\{SOME(2n)\}_{n \geq 1}\) and \(\{SOME(2n-1)\}_{n \geq 1}\) are increasing, and for all \(n\), the sum of odd parts in partitions of \(n\) is at least the sum of even parts. These monotonicity statements refine the interpretation of \(SOME(n)\): the statistic is not only nonnegative in aggregate, but ordered in a parity-sensitive manner across successive arguments.

## 5. The general analogue \(S_{\mathcal P}(n)\)

A general analogue is introduced for arbitrary partition families. Given any family \(\mathcal A(n)\) of partitions of a particular type, and a partition \(\lambda\), define
\[
\omega(\lambda) = \sum_{\text{odd parts of } \lambda} (\text{part}) - \sum_{\text{even parts}} (\text{part}),
\]
and
\[
S_{\mathcal P}(n) = \sum_{\lambda \in \mathcal A(n)} \omega(\lambda).
\]
This extends the \(SOME\)-type statistic from ordinary, distinct, and overpartitions to any chosen family of partitions [2606.31264].

A universal congruence is proved for every partition \(\lambda\) of \(n\):
\[
\omega(\lambda) \equiv n \pmod{4}.
\]
Consequently,
\[
S_{\mathcal P}(4n) \equiv 0 \pmod{4}, \quad S_{\mathcal P}(2n) \equiv 0 \pmod{2}.
\]

The paper also gives mod-\(3\) and general mod-\(k\) criteria. If \(T_r(n)\) denotes the total number of parts congruent to \(r \pmod{6}\) appearing among all partitions in \(\mathcal A(n)\), then
\[
S_{\mathcal P}(n) \equiv T_1(n) + T_2(n) - T_4(n) - T_5(n) \pmod{3},
\]
so that
\[
S_{\mathcal P}(n) \equiv 0 \pmod{3} \iff T_1(n) + T_2(n) \equiv T_4(n) + T_5(n) \pmod{3}.
\]

For general modulus \(k\), let \(L = \mathrm{lcm}(2, k)\), and let \(M_r(n)\) be the number of parts congruent to \(r \bmod L\). Then
\[
S_{\mathcal P}(n) \equiv \sum_{r=0}^{L-1} (-1)^{r+1} r\, M_r(n) \pmod{k},
\]
and hence
\[
S_{\mathcal P}(n) \equiv 0 \pmod{k} \iff \sum_{r=0}^{L-1} (-1)^{r+1} r\, M_r(n) \equiv 0 \pmod{k}.
\]

These statements unify previously known congruences for \(SOME(n)\), including \(SOME(4n) \equiv 0 \pmod{4}\), and extend them to arbitrary partition families.

## 6. Colour partition analogue and divisibility criteria

A colour partition analogue is introduced by allowing each part \(j\) to appear in \(c_j\) different colors. Writing
\[
\Gamma(q) = \prod_{j=1}^\infty \frac{1}{(1 - q^j)^{c_j}},
\]
the generating function is
\[
\sum_{n=0}^\infty S_\mathbf{c}(n) q^n = \Gamma(q) \sum_{j=1}^\infty (-1)^{j+1} j c_j \frac{q^j}{1 - q^j}.
\]
This gives an explicit colour-weighted extension of the \(SOME\)-type statistic [2606.31264].

The main divisibility theorem states:
\[
S_\mathbf{c}(n) \equiv 0 \pmod{k} \text{ for all } n
\quad \text{if and only if} \quad
k \mid j c_j \text{ for all } j \geq 1.
\]

Several corollaries are recorded. For \(k=3\),
\[
S_\mathbf{c}(n) \equiv 0 \pmod{3}
\]
if and only if \(c_j \equiv 0 \pmod{3}\) for all \(j \not\equiv 0 \pmod{3}\). For \(k=4\),
\[
S_\mathbf{c}(n) \equiv 0 \pmod{4}
\]
if and only if
\[
\begin{cases}
4 \mid c_j, & j \equiv 1,3 \pmod{4},\\
2 \mid c_j, & j \equiv 2 \pmod{4},\\
\text{no restriction}, & j \equiv 0 \pmod{4}.
\end{cases}
\]
A further corollary states that if each part in the partition is a multiple of \(k\) and can have any color, then \(S_{ck}(n) \equiv 0 \pmod{k}\).

The colour partition and part-type congruence criteria allow construction of partition families where the \(SOME\)-type statistic is always divisible by a prescribed modulus. In this broader setting, \(SOME(n)\) appears not only as a specific partition statistic, but as the initial case of a divisibility theory spanning ordinary partitions, distinct partitions, overpartitions, arbitrary partition families, and coloured partition structures [2606.31264].

Source: https://www.emergentmind.com/topics/some