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Sign Patterns in a Two Colored Partition Companion series

This paper resolves the Andrews–El Bachraoui sign conjecture for the eta-normalized two-color partition companion series by proving its coefficients oscillate infinitely and provides a combinatorial explanation for their congruence modulo 4 via a Franklin-type involution.

Original authors: Aritram Dhar, Ankush Goswami, Mohit Tripathi

Published 2026-07-14

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CC BY 4.0

Original authors: Aritram Dhar, Ankush Goswami, Mohit Tripathi

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 a giant, magical ledger where numbers are built by stacking blocks. In the world of this paper, mathematicians are looking at a very specific type of block tower called a "two-color partition." Think of it like building a staircase with blocks that come in two flavors: Red and Blue. There's a strict rule for building these towers: the smallest block must be a specific color (say, Red), but every single block above it can be Red, Blue, or even a mix of both.

The authors, Aritram Dhar, Ankush Goswami, and Mohit Tripathi, are investigating the "score" of these towers. They want to know if the total score (which can be positive or negative, like a bank account going up or down) behaves in a predictable way as the towers get taller.

The Great Sign Hunt

First, they looked at a related series of numbers called $c(n)$ . Before this paper, a famous guess (a conjecture) by Andrews and El Bachraoui suggested that as you go further out into the number line, these scores would eventually get incredibly huge in the positive direction and incredibly huge in the negative direction. They thought the numbers would swing wildly like a pendulum that never stops.

The authors didn't just agree with this guess; they proved it was true with a "stronger" version. They showed that for any number you pick—whether it's 2, 100, or 1,000,000—there are infinitely many times the score hits exactly that number (or its negative). It's not just that the numbers get big; it's that they hit every even number on the way up and every even number on the way down, over and over again. They proved this by treating the numbers like a puzzle involving shapes on a grid (specifically, finding solutions to $x^2 + 3y^2 = 4D$ ). By carefully choosing special "prime" building blocks and multiplying them together, they could force the score to land on any target they wanted, proving the pendulum swings forever.

The Magic Mirror Trick

The second part of the paper tackles a different mystery: why do these tower scores follow a specific pattern when divided by 4? The previous researchers had already calculated the pattern using heavy math, but they asked a question: "Can we explain this with a simple, visual trick?"

The authors say "Yes!" and they use a clever magic mirror trick called an "involution." Imagine you have a huge pile of these two-color towers. The authors set up a rule where every single tower can be paired up with a "twin" tower that has the exact opposite score (one is positive, the other negative). When you add them together, they cancel each other out to zero. It's like a dance where every partner cancels the other's move.

However, not everyone finds a partner. Some special towers are their own twins, or they get stuck in a loop where they can't be paired. The authors discovered that the only towers that don't get cancelled out are the ones shaped like perfect triangular staircases (1 block, then 2, then 3, and so on). Because all the other towers cancel out, the final answer is just the sum of these special triangular towers. This visual "dance" perfectly explains why the scores behave the way they do when divided by 4, turning a complex algebra problem into a story about pairing up and cancelling out.

What They Didn't Solve

While they cracked the code for the "modulo 4" pattern (the rule about dividing by 4), they explicitly note that a harder version of the puzzle remains unsolved. There is a conjecture about what happens when you divide by 8. Other mathematicians have recently proved this using different methods, but the authors of this paper did not solve it themselves; they simply pointed out that their "dance" trick hasn't been extended to cover the "divide by 8" case yet. They leave that as a challenge for future explorers.

In short, the paper proves that these number scores swing wildly to infinity in both directions and provides a beautiful, visual explanation for why they follow a specific pattern when divided by 4, using a game of musical chairs where only the triangular towers get to keep their seats.

Technical Summary: Sign Patterns in a Two-Colored Partition Companion Series

Problem Statement
This paper investigates two interrelated problems arising from the work of Andrews and El Bachraoui regarding the two-color partition series $S_1(q)$ and its associated "odd companion" series, $T_o(q)$ . The series $S_1(q)$ enumerates two-color partitions of $n$ into distinct parts where the smallest part is restricted to a specific color, while larger parts may appear in either or both colors. Andrews and El Bachraoui established a complete modulo 4 description for the coefficients $s_1(n)$ and introduced the eta-normalized companion $C(q) = (q;q)_\infty T_o(q)$ , whose coefficients $c(n)$ exhibit complex sign patterns.

The primary objectives of this study are:

To resolve the Andrews–El Bachraoui sign conjecture regarding the asymptotic behavior of $c(n)$ , specifically whether the coefficients assume arbitrarily large positive and negative values.
To provide a combinatorial proof for the modulo 4 congruence of $s_1(n)$ , a problem left open in the original work.

Methodology
The authors employ distinct analytical and combinatorial techniques for each of the two main results.

For the Sign Conjecture (Theorem 3): The authors utilize the signed representation formula for $c(n)$ , which expresses the coefficient as a difference of representation counts of the quadratic form $x^2 + 3y^2 = 4D$ (where $D=6n+7$ ) under specific residue class constraints.
- They construct a sequence of integers $n_k$ by selecting primes $p_j$ of the form $u_j^2 + 3v_j^2$ satisfying specific congruence conditions ( $u_j \equiv 1 \pmod{12}, v_j \equiv 0 \pmod{4}$ ) and small angular constraints.
- Using the multiplication identity for the quadratic form $x^2 + 3y^2$ , they demonstrate that multiplying a base solution by these primes generates $2^k$ distinct positive solutions that all fall into the "plus" sign class $(1, 3) \pmod{(12, 4)}$ .
- They invoke Hecke's prime theorem for Grössencharacters to ensure the existence of infinitely many such primes within the required sectors.
- The negative direction of the limit is derived using the known functional relation $c(25n+28) = -c(n)$ .
For the Combinatorial Proof (Theorem 4): The authors address the modulo 4 congruence for $s_1(n)$ by constructing a sign-reversing involution.
- They first establish that $S_1(q) \equiv F(q) \pmod 4$ , where $F(q)$ is a signed generating function for triples $(a; \lambda, \mu)$ .
- They map these triples to a distinct-part partition model $\rho$ using a bijection $\Phi$ , transforming the problem into finding an involution on partitions into distinct non-negative parts with an even smallest part.
- They utilize the explicit Franklin-type involution $\psi$ constructed by Chen and Liu for distinct-part partitions. This involution preserves the weight and reverses the sign, leaving only specific "fixed points" (staircase partitions) uncancelled.
- By mapping these fixed points back to the original triples, they show that the surviving terms correspond exactly to triangular numbers, thereby proving the congruence.

Key Contributions and Results

Resolution of the Sign Conjecture: The paper proves a strong form of the Andrews–El Bachraoui conjecture. The authors demonstrate that for every integer $k \geq 0$ , there exist infinitely many $n$ such that $c(n) = 2^k$ and infinitely many $n$ such that $c(n) = -2^k$ . Consequently, they establish:
$\limsup_{n \to \infty} c(n) = +\infty \quad \text{and} \quad \liminf_{n \to \infty} c(n) = -\infty.$
This confirms that the coefficients of the eta-normalized companion series oscillate with unbounded amplitude.
Combinatorial Proof of Modulo 4 Congruence: The paper provides the first combinatorial proof (via a signed involution) for the theorem stating that $s_1(n) \equiv (-1)^r \pmod 4$ if $n = r(r+1)/2$ (a triangular number) and $0 \pmod 4$ otherwise. This answers Problem 1 posed by Andrews and El Bachraoui.
Structural Insights: The work clarifies the relationship between partial theta functions, signed partition identities, and Franklin-type cancellations. It explicitly links the parity of $T_o(q)$ to the modulo 8 behavior of $S_1(q)$ away from triangular indices.

Significance and Claims
The authors position this work as a direct advancement on the recent findings of Andrews and El Bachraoui. The significance lies in:

Rigorous Asymptotics: Moving beyond the observation that $c(n)$ changes sign infinitely often to proving that the magnitude of these signs is unbounded.
Combinatorial Explanation: Providing a constructive, bijective explanation for the arithmetic properties of $s_1(n)$ , specifically the modulo 4 behavior, which was previously only proven analytically.
Methodological Application: Successfully adapting the Franklin-type involution framework of Chen and Liu to a two-color partition context, demonstrating the utility of signed involutions in explaining congruences.

The paper concludes by noting that while the modulo 4 case is now combinatorially settled, the modulo 8 congruence conjecture (Conjecture 2 in [3]) remains open for combinatorial proof, and the existence of a rank or crank statistic explaining the modulo 4 result directly is an open question.

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