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Arithmetic Properties Satisfied by a Recent Integer Partition Function of Dombos

This paper investigates the arithmetic properties of a specific integer partition function $dp(n)$, introduced by Dombos, by employing elementary generating function techniques and classical $q$ -series results to establish several congruences, including a family of divisibility results modulo 3 for arguments of the form $3^{2\alpha + 1}n + \frac{7 \cdot 9^\alpha + 1}{4}$ .

Original authors: Robson da Silva, James A. Sellers

Published 2026-06-17

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CC BY 4.0

Original authors: Robson da Silva, James A. Sellers

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 an endless supply of Lego bricks. Your goal is to build a tower that uses exactly a certain number of bricks, say $n$ . In the world of mathematics, this is called a partition. You can stack the bricks in any order, as long as the pieces get smaller or stay the same size as you go up the tower.

For a long time, mathematicians have been fascinated by a specific rule: How many different ways can you build a tower of size $n$ if you follow a very strict set of rules about which bricks you are allowed to use?

The New Rulebook

In this paper, the authors Robson da Silva and James Sellers are looking at a new, quirky rulebook introduced by a researcher named Dombos. The rule for building your tower is simple but picky:

You can only use bricks whose size is a multiple of 4 (like 4, 8, 12...).
OR, you can use bricks that leave a remainder of 1 or 5 when divided by 6 (like 1, 5, 7, 11...).

If you try to use a brick of size 2, 3, or 6, you aren't allowed. The authors call the number of ways to build a tower of size $n$ under these rules $dp(n)$.

The Great Hunt for Patterns

The main goal of this paper is to find hidden patterns in the numbers generated by this rulebook.

Think of it like a lottery. If you look at the winning numbers for a standard lottery, they seem random. But sometimes, if you look at them through a special filter (like only looking at numbers that end in 4), you might discover a secret code: "Every single time the number ends in 4, it is divisible by 5."

The authors are hunting for these "secret codes" (mathematicians call them congruences) for their new partition function $dp(n)$. They want to prove that for certain specific tower sizes, the number of ways to build them is always divisible by a specific number (like 2, 3, 4, or 8), leaving no remainder.

The Discoveries

Using a toolkit of mathematical "magic tricks" (specifically, manipulating complex algebraic formulas called generating functions and q-series), the authors found several of these patterns:

The Evenness Rule: If you try to build a tower of size $6n + 4$ (like 4, 10, 16...), the number of ways to do it is always an even number. It's like saying, "No matter how you try, you can never build this specific tower in an odd number of ways."
The Divisibility by 4 and 8: They found even stricter rules. For certain larger tower sizes (like $18n + 10$ ), the number of ways is always divisible by 4. For others ( $54n + 52$ ), it's always divisible by 8.
The Prime Number Filter: They discovered a rule involving prime numbers (numbers like 17, 23, 41...). If you pick a prime number that fits a specific shape (leaving a remainder of 17 or 23 when divided by 24), you can predict that for a massive range of tower sizes, the number of ways to build them is divisible by 4.
The "Time Travel" Rule: One of their most interesting findings is a relationship between different tower sizes. They proved that the number of ways to build a tower of size $27n + 7$ is exactly the same (in terms of remainders when divided by 3) as building a much smaller tower of size $3n + 1$ . It's as if the pattern for a giant tower is just a "zoomed-in" version of a tiny tower.

How They Did It

The authors didn't just guess these patterns; they built a mathematical machine to prove them.

The Blueprint: They started with a formula (a generating function) that acts like a blueprint, containing all the information about every possible tower size in one giant equation.
The Sifters: They used known mathematical identities (like sifting sand through a sieve) to separate the equation into different parts. They looked specifically for parts of the equation that correspond to the tower sizes they were interested in.
The Proof: By showing that certain parts of the equation always result in numbers that are multiples of 2, 3, or 4, they proved that the number of ways to build those towers must also be multiples of those numbers.

The Big Picture

In short, this paper is a detective story. The "crime" is the apparent randomness of how many ways you can build a tower under Dombos's rules. The "detectives" (the authors) used algebraic tools to uncover a hidden order, proving that for specific tower sizes, the number of construction methods follows strict, predictable rules of divisibility. They didn't just find one rule; they found a whole family of them, including a pattern that holds true for infinitely many cases.

Technical Summary: Arithmetic Properties of a Partition Function of Dombos

Problem Statement
This paper investigates the arithmetic properties of a specific integer partition function, denoted as $dp(n)$, introduced in recent work by Dombos. The function $dp(n)$ counts the number of partitions of an integer $n$ where the parts are restricted to two specific sets: integers divisible by 4, or integers congruent to $\pm 1 \pmod 6$ . The primary objective is to establish Ramanujan-like congruences for $dp(n)$, specifically identifying arithmetic progressions of $n$ for which $dp(n)$ is divisible by small primes (2, 3, and 4).

Methodology
The authors employ elementary generating function manipulations and classical $q$ -series results to derive their findings. The core of the methodology involves:

Generating Function Reformulation: The generating function for $dp(n)$, initially defined as an infinite product involving $q$ -Pochhammer symbols, is rewritten in terms of the standard notation $f_k = (q^k; q^k)_\infty$ . This yields the compact form $G(q) = \frac{f_2 f_3}{f_1 f_4 f_6}$ .
Dissection Techniques: The proofs rely heavily on $k$ $k$ -dissections (specifically 2-dissections and 3-dissections) of various $q$ $q$ -series. The authors utilize well-known identities, including:
- 2-dissections of $f_1^2$ and $1/f_1^2$ (Lemma 2.2).
- 3-dissections of $\frac{f_2}{f_1 f_4}$ , $\frac{f_2}{f_1^2}$ , and $\frac{f_1^2}{f_2}$ (Lemmas 2.3, 2.4, 2.5).
- The 3-dissection of $f_3^3/f_1^3$ and $1/\psi(q)$ (Lemmas 2.6, 2.7).
- Jacobi's identity for $f_1^3$ (Lemma 2.4/14).
Modular Arithmetic: The authors extract coefficients of specific powers of $q$ (corresponding to arithmetic progressions) from the dissection formulas and analyze the resulting expressions modulo 2, 3, 4, and 8.
Induction and Number Theory: For the more complex congruences involving higher powers of 3 and prime moduli, the authors utilize induction on the exponent $\alpha$ and properties of quadratic forms and Legendre symbols.

Key Contributions and Results
The paper establishes several families of congruences for $dp(n)$:

Modulo 2 and 4 Congruences (Theorem 1.2):
- $dp(6n + 4) \equiv 0 \pmod 2$ .
- $dp(18n + 10) \equiv 0 \pmod 4$ .
- $dp(54n + 52) \equiv 0 \pmod 8$ .
  The proof for the modulo 8 result involves showing that the relevant generating function contains no terms of the form $q^{3n+2}$ after expansion.
Prime Modulo Congruences (Theorem 1.3):
For any prime $p \equiv 17$ or $23 \pmod{24}$ , and for all $k, m \ge 0$ where $p \nmid m$ :
$dp\left(\frac{6p^{2k+1}m + 15p^{2k+2} + 1}{4}\right) \equiv 0 \pmod 4.$
This result is derived by analyzing the representation of integers by the quadratic form $6x^2 + y^2$ and utilizing the fact that $\left(\frac{-6}{p}\right) = -1$ for the specified primes.
Modulo 3 Congruences (Theorems 1.4–1.6):
- Base Congruence: $dp(27n + 16) \equiv 0 \pmod 3$ .
- Internal Congruence: $dp(27n + 7) \equiv dp(3n + 1) \pmod 3$ . This establishes a recursive relationship between the partition function at different scales.
- Generalized Congruence: For all $\alpha \ge 1$ and $n \ge 0$ :
  $dp\left(\frac{3^{2\alpha+1}n + 7 \cdot 9^\alpha + 1}{4}\right) \equiv 0 \pmod 3.$
  The generalized result is proven by induction, using the internal congruence (Theorem 1.5) to step from $\alpha$ to $\alpha+1$ .

Significance
The paper positions itself as a direct extension of the classical work of Ramanujan on partition congruences. By proving that the function $dp(n)$, which arises naturally in Dombos's recent work, satisfies similar arithmetic constraints, the authors contribute to the broader study of restricted partition functions. The work demonstrates that despite the specific constraints on the parts (divisibility by 4 or congruence to $\pm 1 \pmod 6$ ), the function retains deep arithmetic structure, exhibiting infinite families of congruences modulo 2, 3, 4, and 8. The results are presented as a "brief note" focused on establishing these specific congruences using classical $q$ -series techniques.