Partition-Theoretic SOME Functions

• Partition-theoretic SOME(n) is defined as the sum of odd parts minus even parts over all partitions, linking combinatorial structure with divisor sums.
• Generating functions for SOME(n), DSOME(n), and overSOME(n) reveal key identities, including connections to pentagonal numbers and classical partition functions.
• Broader generalizations, such as Sₚ(n) and colored partition analogues, yield divisibility criteria and universal congruences across various partition families.

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 (Bardhan et al., 30 Jun 2026).

1. Definitions and generating series

The basic statistics are defined as follows. $DSOME(n)$0 is the sum of all odd parts minus the sum of all even parts over all ordinary partitions of $DSOME(n)$1. $DSOME(n)$2 is the same quantity for distinct partitions, and $DSOME(n)$3 is the corresponding quantity for overpartitions (Bardhan et al., 30 Jun 2026).

For $DSOME(n)$4, the generating function previously established is

$DSOME(n)$5

For $DSOME(n)$6, the generating function is

$DSOME(n)$7

A closed form due to Baruah and Gogoi is also recorded: $DSOME(n)$8 where $DSOME(n)$9.

A generating function for $\overline{SOME}(n)$0 was also previously established, and the 2026 study treats it alongside $\overline{SOME}(n)$1 and $\overline{SOME}(n)$2 as part of a common framework of partition statistics (Bardhan et al., 30 Jun 2026).

2. Structural identity for $\overline{SOME}(n)$3

A central identity links $\overline{SOME}(n)$4 to the partition function $\overline{SOME}(n)$5 and the divisor-sum function $\overline{SOME}(n)$6: $\overline{SOME}(n)$7 Here $\overline{SOME}(n)$8 is the partition function and $\overline{SOME}(n)$9 is the sum of divisors of $SOME(n)$0 (Bardhan et al., 30 Jun 2026).

This identity places $SOME(n)$1 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(n)$2, and later generalizes that pattern to arbitrary partition families through $SOME(n)$3. In that sense, the identity above is both a specific formula for $SOME(n)$4 and a prototype for a broader divisibility theory (Bardhan et al., 30 Jun 2026).

3. $SOME(n)$5, $SOME(n)$6, and interrelations

For $SOME(n)$7, an explicit divisor-sum formula is given: $SOME(n)$8 where $SOME(n)$9 counts distinct partitions and $n$0 is the exponent of $n$1 in $n$2 (Bardhan et al., 30 Jun 2026).

A second identity expresses $n$3 in terms of $n$4: $n$5

The three statistics are further connected by the relation

$n$6

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 $n$7-series and partition-theoretic substrate underlying the three functions.

4. Congruences and monotonicity

The 2026 results establish new congruences for $n$8. For any $n$9,

$SOME(n)$0

and

$SOME(n)$1

The same paper also gives, for any integer $SOME(n)$2, an if and only if criterion characterizing when

$SOME(n)$3

in terms of representation numbers $SOME(n)$4 (Bardhan et al., 30 Jun 2026).

Monotonicity is established for both the ordinary and overpartition versions: $SOME(n)$5 and

$SOME(n)$6

Two consequences are stated explicitly for $SOME(n)$7: both sequences $SOME(n)$8 and $SOME(n)$9 are increasing, and for all $DSOME(n)$0, the sum of odd parts in partitions of $DSOME(n)$1 is at least the sum of even parts. These monotonicity statements refine the interpretation of $DSOME(n)$2: the statistic is not only nonnegative in aggregate, but ordered in a parity-sensitive manner across successive arguments.

5. The general analogue $DSOME(n)$3

A general analogue is introduced for arbitrary partition families. Given any family $DSOME(n)$4 of partitions of a particular type, and a partition $DSOME(n)$5, define

$DSOME(n)$6

and

$DSOME(n)$7

This extends the $DSOME(n)$8-type statistic from ordinary, distinct, and overpartitions to any chosen family of partitions (Bardhan et al., 30 Jun 2026).

A universal congruence is proved for every partition $DSOME(n)$9 of $DSOME(n)$0: $DSOME(n)$1 Consequently,

$DSOME(n)$2

The paper also gives mod-$DSOME(n)$3 and general mod-$DSOME(n)$4 criteria. If $DSOME(n)$5 denotes the total number of parts congruent to $DSOME(n)$6 appearing among all partitions in $DSOME(n)$7, then

$DSOME(n)$8

so that

$DSOME(n)$9

For general modulus $\overline{SOME}(n)$0, let $\overline{SOME}(n)$1, and let $\overline{SOME}(n)$2 be the number of parts congruent to $\overline{SOME}(n)$3. Then

$\overline{SOME}(n)$4

and hence

$\overline{SOME}(n)$5

These statements unify previously known congruences for $\overline{SOME}(n)$6, including $\overline{SOME}(n)$7, and extend them to arbitrary partition families.

6. Colour partition analogue and divisibility criteria

A colour partition analogue is introduced by allowing each part $\overline{SOME}(n)$8 to appear in $\overline{SOME}(n)$9 different colors. Writing

$S_{\mathcal P}(n)$0

the generating function is

$S_{\mathcal P}(n)$1

This gives an explicit colour-weighted extension of the $S_{\mathcal P}(n)$2-type statistic (Bardhan et al., 30 Jun 2026).

The main divisibility theorem states: $S_{\mathcal P}(n)$3

Several corollaries are recorded. For $S_{\mathcal P}(n)$4,

$S_{\mathcal P}(n)$5

if and only if $S_{\mathcal P}(n)$6 for all $S_{\mathcal P}(n)$7. For $S_{\mathcal P}(n)$8,

$S_{\mathcal P}(n)$9

if and only if

$DSOME(n)$00

A further corollary states that if each part in the partition is a multiple of $DSOME(n)$01 and can have any color, then $DSOME(n)$02.

The colour partition and part-type congruence criteria allow construction of partition families where the $DSOME(n)$03-type statistic is always divisible by a prescribed modulus. In this broader setting, $DSOME(n)$04 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 (Bardhan et al., 30 Jun 2026).