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On a two-color partition series and its companions

This paper investigates the two-color distinct-part series $S_1(q)$ and its odd and even companions by determining coefficients modulo 4, establishing Ramanujan-type progressions, providing overpartition interpretations, and proving a quintic self-similarity with specific representation conditions for the odd companion.

Original authors: George E. Andrews, Mohamed El Bachraoui

Published 2026-06-30

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: George E. Andrews, Mohamed El Bachraoui

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 are running a very special bakery. In this bakery, every ingredient (like flour, sugar, or eggs) comes in two colors: Red and Blue.

Usually, when you make a "partition" (which is just a fancy math word for a way of adding up numbers to reach a total), you just care about the numbers. But in this paper, the authors, George Andrews and Mohamed El Bachraoui, are looking at a very specific set of rules for how these colored ingredients can be used to build a "cake" (a number).

Here is the breakdown of their discovery in simple terms:

1. The Main Cake: $S_1(q)$

The authors study a specific type of cake called a "two-color distinct-part series."

The Rules: You can use any number of ingredients (parts), but you can't use the same number twice (they must be "distinct").
The Twist:
- The smallest ingredient in your cake has a strict rule: it must be a specific color (let's say, only Red).
- Every larger ingredient is free to be Red, Blue, or even both at the same time (a "two-color" ingredient).
The Goal: They want to know exactly how many different ways you can bake a cake of a specific size ( $n$ ). They call this number $s_1(n)$ .

The Big Surprise (Theorem 1):
The authors found a magical pattern in the numbers. If you look at the number of ways to bake these cakes, the answer follows a very strict rhythm when divided by 4.

If the size of your cake ( $n$ ) is a "triangular number" (like 1, 3, 6, 10, 15... which are formed by stacking dots in a triangle), the number of ways to bake it is odd.
If the size is not a triangular number, the number of ways is even (specifically, it's a multiple of 4).

It's like a light switch: Triangular numbers turn the light ON (odd), and everything else keeps it OFF (even).

2. The Siblings: To and Te

The main cake ( $S_1$ ) is actually made of two smaller "sibling" cakes, which the authors call the Odd Companion ( $T_o$ ) and the Even Companion ( $T_e$ ).

The Even Companion ( $T_e$ ): This one counts a different kind of cake called an "overpartition." Imagine a cake where the biggest ingredient can be "highlighted" or "overlined" (like putting a star on top). The authors found that this series counts cakes where the biggest ingredient is even, and only even ingredients can be highlighted. They proved that this number is always exactly double the number of a specific type of standard cake.
The Odd Companion ( $T_o$ ): This is the more mysterious sibling. To understand it, the authors "normalized" it (multiplied it by a special factor) to create a new series called $C(q)$ .

3. The Mystery Box: The $C(q)$ Series

The series $C(q)$ is the star of the show for the second half of the paper. It behaves like a chaotic but predictable machine.

The Quintic Self-Similarity: The authors discovered that if you look at the numbers in this series, they repeat themselves in a pattern related to the number 25. Specifically, if you look at the number for a cake size of $25n + 28$ $25 n + 28$ , it is exactly the negative of the number for size $n$ $n$ .
- Analogy: Imagine a mirror that flips the image upside down and shrinks it. If you look at a cake of size 28, you see the opposite of size 0. If you look at size 53 ( $25\times1 + 28$ ), you see the opposite of size 1. This creates an infinite tower of numbers that flip between positive and negative forever.
The Shape Constraint (Theorem 3): The authors found that this series only "bakes" cakes of certain sizes. It refuses to bake a cake of size $n$ $n$ unless the number $24n + 28$ $24 n + 28$ can be drawn as a specific geometric shape: a square plus three times another square ( $x^2 + 3y^2$ $x^{2} + 3 y^{2}$ ).
- If a number cannot be drawn in this shape, the answer is exactly zero. It's as if the oven has a sensor that only turns on for specific geometric shapes.

4. Why This Matters (In Math Terms)

The paper doesn't talk about real-world applications like engineering or medicine. Instead, it solves puzzles about the hidden structure of numbers.

They proved congruences: Rules that tell us when a number is divisible by 4 or 2.
They found vanishing progressions: Long lists of numbers where the answer is guaranteed to be zero.
They connected these numbers to geometry (shapes like $x^2 + 3y^2$ ) and modular forms (advanced symmetry patterns in math).

Summary

Think of this paper as a detective story about a bakery with colored ingredients.

The authors found that the number of ways to bake a specific type of cake is either odd or even based on whether the cake size is a "triangle number."
They broke this down into two parts (Odd and Even).
The "Odd" part, when cleaned up, revealed a series that flips signs infinitely and only exists for numbers that fit a specific geometric puzzle.

The paper provides the exact formulas and proofs for these patterns, showing that even in the chaotic world of adding numbers, there are deep, rigid, and beautiful rules waiting to be discovered.

Technical Summary: On a Two-Color Partition Series and Its Companions

Problem Statement
This paper investigates the arithmetic and combinatorial properties of a specific two-color distinct-part partition series, denoted $S_1(q)$ , and its associated "odd" and "even" companion series, $T_o(q)$ and $T_e(q)$ . The series $S_1(q)$ serves as the generating function for two-color partitions into distinct parts where the smallest part is restricted to a single prescribed color, while larger parts may appear in either or both colors. Equivalently, $S_1(q)$ is the generating function $v_d(q)$ for strictly concave compositions. The authors aim to determine the coefficients of these series modulo 4, establish congruence relations (Ramanujan-type progressions), provide combinatorial interpretations for the coefficients, and analyze the arithmetic support of an eta-normalized version of the odd companion.

Methodology
The authors employ a combination of $q$ -series analysis and partition-theoretic combinatorics.

Analytic Tools: The proofs rely heavily on standard identities from the theory of basic hypergeometric series, including the $q$ -binomial theorem, Euler's identities and expansions, Gauss' sum, Jacobi's triple product identity, and Heine's transformation. A central technique involves decomposing series into even and odd components and utilizing residue extractions modulo 5 to establish self-similarity properties.
Combinatorial Interpretations: The paper constructs direct interpretations of the coefficients of $T_e(q)$ in terms of overpartitions (partitions where the first occurrence of a part may be overlined) and ordinary partitions with specific constraints on missing parts.
Modular Forms and Quadratic Forms: The support of the coefficients of the eta-normalized series is analyzed using properties of binary quadratic forms, specifically $x^2 + 3y^2$ , and Legendre symbols to derive vanishing conditions for coefficients.

Key Contributions and Results

Modulo 4 Analysis of $S_1(q)$ :
The authors provide a complete description of the coefficients $s_1(n)$ of $S_1(q)$ modulo 4.
- Theorem 1: $s_1(n) \equiv (-1)^r \pmod 4$ if $n$ is a triangular number ( $n = r(r+1)/2$ ), and $s_1(n) \equiv 0 \pmod 4$ otherwise.
- Corollaries: This result yields a complete criterion for Ramanujan-type progressions. Specifically, $s_1(An+B) \equiv 0 \pmod 4$ if and only if $8B+1$ is not a quadratic residue modulo $8A$ . For odd primes $p$ , this reduces to the condition that the Legendre symbol $\left(\frac{8B+1}{p}\right) = -1$ .
Combinatorial Interpretations of the Even Companion $T_e(q)$ :
- The coefficients $t_e(N)$ of $T_e(q)$ are shown to count overpartitions of $N$ where the largest part is even, only even parts may be overlined, and odd parts are distinct.
- Theorem 4: The authors prove that $t_e(N)$ $t_{e} (N)$ is even for all $N \ge 1$ $N \geq 1$ . Furthermore, $t_e(N)/2$ $t_{e} (N) /2$ counts two distinct families of partitions:
  1. Ordinary partitions of $N$ where the first missing part among integers not congruent to $1 \pmod 4$ is congruent to $3 \pmod 4$ .
  2. Overpartitions of $N$ with the same constraints as above but with the largest part not overlined.
- This establishes the equality $a_e(N) = b_e(N) = t_e(N)/2$ .
Arithmetic Properties of the Odd Companion $C(q)$ :
The paper studies the eta-normalized series $C(q) = (q; q)_\infty T_o(q)$ with coefficients $c(n)$ .
- Quintic Self-Similarity (Theorem 2): The coefficients satisfy the relation $c(25n + 28) = -c(n)$ . This implies that $c(n)$ changes sign infinitely often and assumes both positive and negative values infinitely often.
- Vanishing Progressions: The coefficients vanish for specific arithmetic progressions: $c(25n + 8) = c(25n + 13) = c(25n + 18) = c(25n + 23) = 0$ .
- Support Constraint (Theorem 3): If $c(n) \neq 0$ , then $24n + 28$ must be representable by the binary quadratic form $x^2 + 3y^2$ . Consequently, if a prime $p \equiv 2 \pmod 3$ divides $6n+7$ to an odd order, then $c(n) = 0$ . This leads to exact vanishing progressions modulo $p^2$ for such primes.
Connection to OEIS A344650:
The paper compares $T_o(q)$ with the generating function for strict partitions of $2n$ into an odd number of parts (OEIS A344650). While the series agree for low-order terms, they diverge at $q^{13}$ , with the authors providing explicit generating function formulas for both.

Significance and Claims
The paper claims to offer a "complete description modulo 4" for the sequence $s_1(n)$ , refining previous parity results found in Andrews' work on concave compositions. It provides the first direct overpartition interpretation for the coefficients of the even companion series $T_e(q)$ and demonstrates that two seemingly different partition families are counted by the same sequence.

For the odd companion, the authors establish a "quintic self-similarity" and derive "exact vanishing relations" that constrain the support of the coefficients to numbers related to the quadratic form $x^2 + 3y^2$ . The paper concludes by noting that while the coefficients of $C(q)$ are sparse, they are unbounded in magnitude (giving examples of values like $\pm 3, \pm 4, \pm 5$ ), and it posits two conjectures: that the coefficients are unbounded in both positive and negative directions, and that a specific 2-adic divisibility condition holds for the coefficients of $T_o(q)$ based on the prime factorization of $8n+9$ .

The authors explicitly frame the work as a study of the arithmetic structure of these specific $q$ -series and their combinatorial counterparts, without proposing new applications beyond the theoretical framework of partition theory and $q$ -series identities.

1. The Main Cake: S1(q)S_1(q)S1​(q)