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Extending recent work of Nath, Saikia, and Sarma on $k$ -tuple $\ell$ -regular partitions

This paper confirms a conjecture by Nath, Saikia, and Sarma regarding infinite congruences for $T_{2,3}(n)$ modulo 6 and establishes new families of congruences for $k$ -tuple $\ell$ -regular partitions using elementary $q$ -series techniques.

Original authors: Bishnu Paudel, James A. Sellers, Haiyang Wang

Published 2026-06-29

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CC BY 4.0

Original authors: Bishnu Paudel, James A. Sellers, Haiyang Wang

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 numbered blocks. Your job is to build towers using these blocks. In the world of math, this is called a partition. You can stack the blocks in any way you like, as long as the numbers on the blocks go down (or stay the same) as you go up the tower, and the total sum of the numbers equals a specific target number, $n$ .

For a long time, mathematicians have been fascinated by a specific rule: What if you aren't allowed to use any blocks that are multiples of a certain number?

For example, if the rule says "No multiples of 2," you can't use the blocks labeled 2, 4, 6, 8, etc. You can only use 1, 3, 5, 7, and so on. This is what mathematicians call an $\ell$ -regular partition (where $\ell$ is the forbidden number).

The New Twist: The "Tuple" Tower

In this paper, the authors (Paudel, Sellers, and Wang) aren't just looking at one single tower. They are looking at groups of towers that must be built together.

Imagine you are asked to build a set of 3 towers (a "3-tuple") that all share the same total number of blocks.

Tower A uses some blocks.
Tower B uses some blocks.
Tower C uses some blocks.
The sum of blocks in all three towers combined must equal $n$ .

The rule is strict: Every single block in every single tower must follow the "no multiples of $\ell$ " rule. The authors are counting how many different ways you can build these specific sets of towers. They call this count $T_{\ell,k}(n)$ .

The Mystery of the Patterns

Mathematicians love finding hidden patterns in numbers. A famous mathematician named Ramanujan discovered long ago that if you look at the total number of ways to build any tower (without the "no multiples" rule), the numbers follow a very specific rhythm: every 5th number in a certain sequence is divisible by 5, every 7th is divisible by 7, and so on.

Recently, a team of researchers (Nath, Saikia, and Sarma) looked at the "Tuple Towers" described above. They found some cool patterns (congruences) for specific cases, but they hit a wall. They noticed a pattern that seemed to go on forever for a specific case (using 2 as the forbidden number and 3 towers), but they couldn't prove it was true for every number. They threw out a conjecture (a guess based on strong evidence): "We bet this pattern holds true for an infinite number of cases."

What This Paper Does

The authors of this paper, Paudel, Sellers, and Wang, stepped in to solve that mystery.

They Proved the Conjecture: They took the guess Nath, Saikia, and Sarma made and proved it was 100% correct. They showed that for a specific type of "Tuple Tower," the number of ways to build it is always divisible by 6 (and actually by 24!) for an infinite list of numbers.
They Found Even More Patterns: They didn't stop at just proving the guess. They used some clever, basic math tricks (which they call "elementary techniques," meaning they didn't need super-complex machinery) to discover new families of patterns.
- They found that for many different "forbidden numbers" and "tower counts," the results are always divisible by 8, and sometimes by 24.
- They showed that these patterns work for a wide variety of prime numbers (like 3, 5, 7, etc.).

How They Did It (The "Magic" Tricks)

You don't need to know the math to understand the approach. Think of it like this:

The Generating Function: The authors use a special "recipe" (a mathematical formula) that, when you expand it, lists out all the possible ways to build the towers. It's like a machine that spits out the answer for every number $n$ at once.
The Filter: They apply simple rules to this machine. They look at the formula and say, "If we change the numbers slightly, certain parts of the recipe cancel each other out."
The Result: When those parts cancel out, the remaining numbers are always multiples of 8 or 24. It's like finding that no matter how you arrange the blocks, if you follow a specific path, you will always end up with a pile of blocks that can be perfectly divided into groups of 8.

The Bottom Line

This paper is a victory for pattern hunters.

Before: Mathematicians had a strong hunch that a specific pattern existed for "3-tower sets with no multiples of 2," but they couldn't prove it for every single case.
Now: They have a solid proof. Not only is the hunch correct, but there are actually many more similar patterns waiting to be found for other combinations of towers and rules.

The authors didn't just confirm a guess; they opened the door to a whole new hallway of mathematical patterns, showing that these "Tuple Towers" have a very deep, rhythmic structure that repeats forever.

Technical Summary: Extending Recent Work of Nath, Saikia, and Sarma on $k$ -Tuple $\ell$ -Regular Partitions

Problem Statement
The paper investigates the arithmetic properties of $T_{\ell,k}(n)$ , the number of $\ell$ -regular $k$ -tuple partitions of a non-negative integer $n$ . An $\ell$ -regular partition is defined as a partition where no part is divisible by $\ell$ . A $k$ -tuple partition consists of an ordered sequence of $k$ partitions $(\lambda_1, \dots, \lambda_k)$ summing to $n$ , where each $\lambda_i$ is an $\ell$ -regular partition of its respective sum. The authors focus specifically on the case where $\ell=2$ and $k=3$ , denoted as $T_2(n) := T_{2,3}(n)$ .

The work is motivated by a recent study by Nath, Saikia, and Sarma [12], who established several families of congruences for $T_2(n)$ modulo 24 and 6. In their concluding remarks, Nath, Saikia, and Sarma conjectured that $T_2(n)$ satisfies an infinite set of congruences modulo 6 under specific conditions involving primes $p$ and integers $t$ .

Methodology
The authors employ entirely elementary techniques based on $q$ -series manipulations and generating functions. They do not utilize advanced modular forms or complex analytic methods. The core of their approach involves:

Generating Functions: Utilizing the generating function for $T_{\ell,k}(n)$ , given by $\sum T_{\ell,k}(n)q^n = f_\ell^k / f_1^k$ , where $f_k = (q^k; q^k)_\infty$ .
Classical Identities: Leveraging standard $q$ -series identities, including Euler's Pentagonal Number Theorem (Lemma 2.1), Jacobi's Triple Product identity variants (Lemmas 2.2, 2.3), and specific expansions for powers of $f_1$ and $f_2$ (Lemmas 2.4–2.6).
Congruence Properties: Applying the Binomial Theorem to derive congruences for $q$ -Pochhammer symbols modulo prime powers (Lemmas 2.7, 2.8). Specifically, they use the property that $f_{p^k} \equiv f_p^{k-1} \pmod{p^k}$ to isolate coefficients in the generating function.
Quadratic Form Analysis: To prove congruences modulo 8, the authors analyze the representation of specific integers as sums of quadratic forms (e.g., $x^2 + 2y^2$ or $x^2 + y^2$ ). They utilize Legendre symbols to demonstrate that certain integers cannot be represented by these forms, thereby proving that the corresponding coefficients in the generating function vanish modulo 8.

Key Contributions and Results

Confirmation of Conjecture 1.1:
The authors prove a stronger version of the conjecture proposed by Nath, Saikia, and Sarma.
- Theorem 1.2: For any positive integer $t$ with $\gcd(t, 6)=1$ and $N \in \{33, 57\}$ , the congruence $T_2\left(\frac{9n + Nt^2 - 1}{8}\right) \equiv 0 \pmod{24}$ holds for all $n \ge 0$ .
- Corollary 1.3: This result directly confirms Conjecture 1.1, establishing the infinite family of congruences modulo 6 (and actually modulo 24) for the specified parameters.
New Infinite Families of Congruences:
The paper establishes a new, extremely elementary proof for an infinite family of congruences modulo 8 and 24.
- Theorem 1.4: Let $p$ be a prime such that $p \equiv 3, 5,$ or $7 \pmod 8$ . For $n, \alpha \ge 0$ with $p \nmid n$ , the following holds:
  $T_2\left(\frac{p^{2\alpha+1}n + p^{2\alpha+2} - 1}{8}\right) \equiv 0 \pmod 8.$
- Furthermore, if $3 \nmid n$ , the congruence strengthens to:
  $T_2\left(\frac{p^{2\alpha+1}n + p^{2\alpha+2} - 1}{8}\right) \equiv 0 \pmod{24}.$
- Corollary 1.5: A specific case is derived where the modulus 24 congruence holds for $p \neq 3$ .
Generalization of Previous Results:
The authors show that the congruence previously established by Nath, Saikia, and Sarma (Equation 3 in the paper) holds modulo 24 (not just modulo 6) and extends to primes $p \equiv 3 \pmod 8$ .

Significance and Claims
The paper claims significance primarily in the confirmation of the conjecture by Nath, Saikia, and Sarma and the extension of these results to a broader class of primes and moduli. The authors emphasize that their proofs are "extremely elementary," relying solely on generating function manipulations and classical $q$ -series results rather than heavy machinery.

In the "Closing Thoughts" section, the authors note that their computations suggest further divisibility properties, specifically that $T_2(n)$ appears to be divisible by 32 for certain arithmetic progressions (e.g., $T_2(25n+8)$ , $T_2(49n+13)$ , etc.) and that $T_2(7n+6)$ may be divisible by 29. However, they explicitly state that these observations are left as open questions for interested readers and do not claim to have proven these specific higher-modulus congruences in this work. The paper concludes by framing these as potential directions for future investigation rather than established theorems.