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Some new congruences and identities for $SOME(n)$ , $DSOME(n)$ , $\overline{SOME}(n)$ functions and analogues

This paper establishes new identities, congruences, monotonicity results, and divisibility properties for the $SOME(n)$ , $DSOME(n)$ , and $\overline{SOME}(n)$ partition functions, while also introducing and analyzing general and colored partition analogues of these functions.

Original authors: Gaurab Bardhan, Nipen Saikia

Published 2026-07-01

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Gaurab Bardhan, Nipen Saikia

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a giant bag of Lego bricks. Each brick has a size (1, 2, 3, etc.). A "partition" of a number is just a way of building a tower that uses exactly that many total bricks. For example, if your target number is 4, you could build a tower with four 1s, or two 2s, or a 3 and a 1, and so on.

Mathematicians love counting how many different towers they can build for any number. But this paper isn't just about counting the towers; it's about weighing them in a very specific, quirky way.

Here is the breakdown of what the authors, Gaurab Bardhan and Nipen Saikia, are doing, explained simply:

1. The "Odd vs. Even" Scale

The main characters in this story are three functions: SOME(n), DSOME(n), and SOME(n) (with a bar over it).

Think of these functions as a special scale.

SOME(n): You look at every possible tower you can build with $n$ bricks. For every tower, you add up the sizes of all the odd-numbered bricks (1, 3, 5...) and subtract the sizes of all the even-numbered bricks (2, 4, 6...). Then, you add up these results for all the towers.
DSOME(n): This is the same game, but you only build towers where no two bricks are the same size (a "distinct" partition).
SOME(n) (with a bar): This is the "Overpartition" version. Imagine some of your bricks have a special "highlighter" mark on them. You can use a regular brick or a highlighted brick of the same size, but you can't use the same highlighted brick twice. This function does the odd-minus-even calculation for these special highlighted towers.

The Big Question: The authors are asking, "When we do this weird math, do the results follow any hidden patterns?"

2. The "Magic Rules" (Congruences)

In math, a "congruence" is like a secret code that says, "If you divide this number by 4 (or 5, or 8), the remainder is always 0."

The authors found several of these secret codes:

The "Divisible by 4" Rule: If you take a number like 4, 8, 12, etc., and run it through the SOME or DSOME functions, the result is always perfectly divisible by 4. It's like the universe insists that these specific numbers always come out even in pairs of twos.
The "Perfect Square" Rule: If your number $n$ is a perfect square (like 1, 4, 9, 16) and it's odd, the result is always 2 more than a multiple of 4. If it's not a perfect square, the result is a multiple of 4.
The "Divisible by 5" Rule: They confirmed that for certain numbers (like 2, 7, 12... which are $5n+2$ ), the result is always divisible by 5.

They didn't just find these rules; they proved them using complex algebraic formulas (generating functions), which are like blueprints that describe the entire infinite collection of towers at once.

3. The "Monotonicity" (The Staircase Effect)

One of the most interesting findings is about growth.
Imagine you have a staircase. The authors proved that if you look at the "Odd minus Even" score for a number $n$ , and then look at the score for $n-2$ (two steps back), the score for $n$ is almost always higher or equal.

The Analogy: It's like climbing a hill. As you go higher (larger numbers), the "Odd minus Even" score generally goes up. It doesn't wiggle back and forth randomly; it has a steady upward trend for even numbers and a steady upward trend for odd numbers separately.
The Takeaway: This means that for any number $n$ , the sum of all the odd parts in all possible towers is greater than or equal to the sum of all the even parts. The "Odd" team always wins or ties the "Even" team.

4. The "General Analogue" (The Universal Rule)

The authors didn't stop at just these three specific functions. They created a universal version called SP(n).

The Metaphor: Imagine you have a rulebook for building towers. Maybe you only allow prime-sized bricks, or maybe you only allow bricks that are multiples of 3. SP(n) is a function that works for any rulebook you can invent.
The Discovery: They proved that no matter what rulebook you use, if you take the "Odd minus Even" score for a tower of size $4n$ , it will always be divisible by 4. It's a universal law that holds true regardless of how you restrict your Lego building.

5. The "Colorful" Version

Finally, they imagined a world where every brick comes in different colors.

If you have a brick of size 3, maybe it can be Red, Blue, or Green.
They created a function called Sc(n) to handle these "colored partitions."
The Result: They found a condition where, if the number of colors you allow for each brick size follows a specific pattern, the final "Odd minus Even" score will always be divisible by a specific number (like 3 or 4). It's like saying, "If you paint your bricks in groups of 3, the math will always balance out perfectly."

Summary

In plain English, this paper is a detective story about numbers. The authors looked at a very specific way of counting and weighing partitions (ways to break down numbers). They discovered that despite the chaotic nature of how numbers can be broken down, there are rigid, predictable patterns (divisibility rules) and a steady upward trend (monotonicity) in the results. They also showed that these patterns aren't just flukes for one specific type of number, but apply to a whole family of mathematical "games" involving partitions.

Technical Summary: New Congruences and Identities for SOME(n), DSOME(n), and Analogues

Problem Statement and Context
This paper investigates arithmetic properties of partition-related functions introduced by Andrews and Dastidar (2026) and Gireesh and Hemanthkumar. Specifically, it focuses on:

SOME(n): The sum of all odd parts minus the sum of all even parts across all ordinary partitions of a positive integer $n$ .
DSOME(n): The analogous difference calculated over partitions of $n$ into distinct parts.
$\overline{\text{SOME}}(n)$ : An overpartition analogue of SOME(n).

While previous works established generating functions and congruences modulo 4, 5, and powers of 2 for these functions, this paper aims to derive new identities, establish monotonicity results, and generalize these concepts to arbitrary families of partitions and color partitions.

Methodology
The authors employ standard techniques from the theory of $q$ -series and partition analysis. Key methodological tools include:

Generating Functions: Manipulation of infinite products and series, utilizing standard notation such as $(a; q)_\infty$ and $g_t = (q^t; q^t)_\infty$ .
Ramanujan's Theta Functions: Utilization of $\phi(q)$ , $\psi(q)$ , and $f(-q)$ to transform and analyze series representations.
Lambert Series and Divisor Sums: Expansion of series involving $\sigma(n)$ (sum of divisors) and $r_k(n)$ (representations as sums of squares).
Truncated Pentagonal Number Theorem: Application of results by Andrews and Merca to prove monotonicity properties.
Logarithmic Differentiation: Used to derive generating functions for generalized functions.
Möbius Inversion: Applied in the context of color partitions to establish divisibility conditions.

Key Contributions and Results

1. Identities and Congruences for SOME(n) and DSOME(n)

New Identity (Theorem 3.1): The authors prove a recurrence relation linking SOME(n) to the partition function $p(n)$ and the divisor sum $\sigma(n)$ :
$\text{SOME}(n) + 4 \sum_{i=1}^{\lfloor n/2 \rfloor} p(n-2i)\sigma(i) = n p(n)$
Congruences Modulo 4 and 8 (Theorems 3.2 & 3.3): It is established that $\text{SOME}(n) \equiv 2 \pmod 4$ (and $\pmod 8$ ) if $n$ is an odd perfect square, and $0$ otherwise.
General Congruence (Theorem 3.4): A condition is provided for $\text{SOME}(n) \equiv 0 \pmod{2^{\alpha+2}}$ involving sums of representations of integers as sums of squares.
DSOME(n) Identities (Theorems 3.5 & 3.6): The paper derives an explicit formula for $\text{DSOME}(n)$ involving $pd(n)$ (distinct partitions) and $\sigma(n)$ , and establishes a relation between $\text{DSOME}(n)$ and $\text{SOME}(n)$ using Euler's pentagonal number theorem.

2. Monotonicity Results

Theorems 3.8 and 3.12: Using the truncated pentagonal number theorem, the authors prove that $\text{SOME}(n) \ge \text{SOME}(n-2)$ for $n \ge 2$ .
Corollaries: This implies that the sequences $\{\text{SOME}(2n)\}$ and $\{\text{SOME}(2n-1)\}$ are monotonically increasing. Consequently, the sum of all odd parts in partitions of $n$ is greater than or equal to the sum of all even parts. Similar monotonicity results are proven for the overpartition analogue $\overline{\text{SOME}}(n)$ .

3. General Analogue SP(n)

Definition: The paper defines $SP(n)$ as the sum of (odd parts - even parts) over any arbitrary family of partitions $P(n)$ .
Divisibility Properties:
- Theorem 4.1: For any partition $\lambda$ of $n$ , the difference between the sum of odd and even parts satisfies $\omega(\lambda) \equiv n \pmod 4$ .
- Theorem 4.3 & 4.4: The authors establish necessary and sufficient conditions for $SP(n) \equiv 0 \pmod k$ based on the counts of parts congruent to specific residues modulo $L = \text{lcm}(2, k)$ . Specifically, $SP(n) \equiv 0 \pmod 3$ if and only if the count of parts congruent to 1 or 2 (mod 6) equals the count of parts congruent to 4 or 5 (mod 6).

4. Color Partition Analogue $S_c(n)$

Definition: The paper introduces $S_c(n)$ for partitions where parts of size $j$ appear in $c_j$ different colors.
Generating Function (Theorem 5.1): A closed-form generating function is derived using logarithmic differentiation.
Divisibility Condition (Theorem 5.2): A necessary and sufficient condition is proven for $S_c(n) \equiv 0 \pmod k$ : this holds if and only if $k$ divides $j c_j$ for all $j \ge 1$ .
Corollaries: Specific divisibility rules are derived for moduli 3 and 4, and for cases where parts are restricted to multiples of $k$ .

Significance and Claims
The paper claims to extend the arithmetic theory of partition difference functions by:

Providing new identities that relate these functions to classical partition and divisor functions.
Establishing monotonicity, a property not previously highlighted for these specific difference functions.
Generalizing the framework to arbitrary partition families ($SP(n)$) and color partitions ( $S_c(n)$ ), thereby unifying various divisibility phenomena under a single theoretical structure.
Proving specific congruences modulo powers of 2 and other integers, refining the understanding of the arithmetic behavior of these functions.

The authors maintain a modest scope, focusing strictly on theoretical derivations and proofs within the realm of integer partitions, without proposing experimental applications or future extensions beyond the defined mathematical structures.