🔢 mathematics

Analytic proofs of Andrews-Bachraoui identities related to two-color partitions with evens in one color

This paper provides analytic proofs for previously open $q$ -series identities related to the two-color partition functions $F(n)$ and $H(n)$ , and establishes new congruences for the restricted partition functions $F_0(n)$ and $F_1(n)$ modulo 2, 4, and 8.

Original authors: Gaurab Bardhan, Nipen Saikia

Published 2026-07-09

📖 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 are running a massive, infinite warehouse of building blocks. These blocks come in different sizes (1, 2, 3, 4, etc.) and you want to build towers that add up to a specific total height, say 6. In the world of mathematics, this is called a partition.

This paper is about a very specific, colorful version of this game, invented by mathematicians Andrews and Bachraoui, and now analyzed in detail by the authors of this paper, Gaurab Bardhan and Nipen Saikia.

The Game: Two-Color Blocks with a Twist

In this specific warehouse, the rules are a bit quirky:

Even-sized blocks (2, 4, 6...) can only be Blue.
Odd-sized blocks (1, 3, 5...) can be either Red or Blue.

So, if you want to build a tower of height 3, you could use:

Three Blue 1s.
One Red 1 and two Blue 1s.
One Blue 3.
One Red 3.
And so on.

The authors are interested in counting how many different ways you can build these towers. But they don't just want the total count; they want to sort the towers into special categories based on a "Red Block Count."

The Categories: The Red Block Parity

The authors define two main groups of towers:

The "Even Red" Group ( $F_0$ ): Towers where the number of Red blocks is an even number (0, 2, 4...).
The "Odd Red" Group ( $F_1$ ): Towers where the number of Red blocks is an odd number (1, 3, 5...).

They also look at a stricter rule called $H(n)$ , where you are not allowed to use the same block size more than once within the same color. It's like saying, "You can have a Red 1 and a Blue 1, but you can't have two Red 1s."

What Did the Authors Actually Do?

The paper has two main missions, which the authors tackle using a powerful mathematical toolkit called $q$ -series (think of these as complex algebraic recipes that generate infinite lists of numbers).

Mission 1: Solving the "Open Problems" (The Recipes)

In a previous paper, Andrews and Bachraoui wrote down some very complicated-looking formulas (equations) that they believed described the behavior of these towers. However, they couldn't prove why these formulas worked; they were just "open problems" (unsolved mysteries).

Bardhan and Saikia stepped in and provided the analytic proofs.

The Analogy: Imagine someone gave you a magic recipe for a cake that tastes perfect, but they didn't explain the chemistry behind why the ingredients mix that way. These authors wrote the "chemistry textbook" that proves the recipe works every single time.
They proved two specific, complex equations (labeled 1.1 and 1.2 in the paper) that link the number of these colorful partitions to infinite mathematical products.

Mission 2: Finding the "Hidden Patterns" (The Congruences)

The second part of the paper is like a detective story looking for patterns in the numbers. The authors asked: "If I build a tower of a certain size, can I predict whether the number of ways to build it is even or odd, or divisible by 4 or 8?"

They discovered some surprising rules:

The Square Rule: If the total height of the tower ( $n$ ) is a perfect square (like 1, 4, 9, 16) or twice a perfect square (like 2, 8, 18), the number of ways to build it is odd.
The "Otherwise" Rule: If the height is not one of those special numbers, the number of ways to build it is even.

They went even deeper, finding rules for when the numbers are divisible by 4 or 8. For example, they found that for certain heights (like $4n+3$ ), the number of ways to build the tower is always divisible by 4.

The "Correction" Note

The authors also took a moment to fix a small mistake in the original work by Andrews and Bachraoui. The original paper had a few signs flipped (plus instead of minus) in their formulas. The authors of this paper corrected these signs and provided the right list of partitions for a specific example (height 6), showing exactly which towers belong to the "Even Red" group and which belong to the "Odd Red" group.

Summary

In simple terms, this paper is a mathematical proof and pattern-finding exercise.

It verified that two complex formulas describing colorful block towers are correct.
It corrected a few typos in the original formulas.
It discovered that the number of ways to build these towers follows strict rules based on whether the tower's height is a square number or not, and whether the count is divisible by 2, 4, or 8.

The paper stays strictly within the realm of pure mathematics (number theory). It does not claim these findings will be used for engineering, medicine, or computer science; it simply solves the puzzle of how these specific numbers behave.

Technical Summary: Analytic Proofs of Andrews-Bachraoui Identities and Congruences for Two-Color Partitions

Problem Statement
This paper addresses open problems and corrections regarding the two-color partition function $F(n)$ , introduced by Andrews and Bachraoui. In this partition model, odd parts may appear in two colors (red and blue), while even parts appear in a single color (blue). The authors focus on three specific restricted versions of $F(n)$ :

$F_0(n)$ : The number of partitions where the count of red odd parts is even.
$F_1(n)$ : The number of partitions where the count of red odd parts is odd.
$H(n)$ : The number of partitions where parts of the same color do not repeat.

The primary objectives are to provide analytic proofs for two $q$ -series identities connected to $F(n)$ and $H(n)$ that were previously listed as open problems, and to establish new congruence properties for $F_0(n)$ and $F_1(n)$ modulo 2, 4, and 8. Additionally, the paper corrects errors found in the generating functions presented in the original work by Andrews and Bachraoui.

Methodology
The authors employ standard techniques from the theory of partitions and $q$ -series. The methodology includes:

Generating Functions: Utilizing the generating functions for $F_0(n)$ and $F_1(n)$ , which are expressed in terms of infinite $q$ -products involving $(q; q)_\infty$ . The paper notes a correction to the signs in the numerators of these products found in previous literature.
Telescoping Sums: For the $q$ -series identities, the authors define specific sequences ( $A_n, B_n, C_n, D_n$ ) and demonstrate that the terms in the series represent telescoping differences (e.g., $A_n - A_{n-1}$ ). By summing these differences and taking the limit as $n \to \infty$ , they derive the closed-form identities.
Theta Functions and Dissections: To prove congruences, the authors utilize Ramanujan's general theta function $f(\alpha, \beta)$ and its special cases $\phi(q)$ and $\psi(q)$ . They apply 2-dissection formulas and modular arithmetic properties of $q$ -products (specifically modulo 2, 4, 8, and 16) to extract coefficients.
Number Theoretic Functions: The proofs for congruences modulo 8 involve the sum-of-divisors function $\sigma(n)$ , the number of representations of an integer as a sum of two squares $r_2(n)$ , and the number of representations as $u^2 + 2v^2$ . The authors relate these to Jacobi symbols and divisor sums to characterize the behavior of the partition functions.

Key Contributions and Results

Correction of Generating Functions: The paper identifies and corrects errors in the generating functions for $F_0(n)$ and $F_1(n)$ as presented in Andrews and Bachraoui's original work, specifically regarding the signs in the $q$ -product numerators.
Proof of Open Identities: The authors provide rigorous analytic proofs for two identities (labeled 1.1 and 1.2 in the text) involving infinite series and products related to $F(n)$ and $H(n)$ . These identities were previously unproven in the literature.
Congruences Modulo 2:
- It is proven that $F_i(n) \equiv 1 \pmod 2$ if and only if $n$ is a perfect square or twice a perfect square; otherwise, $F_i(n) \equiv 0 \pmod 2$ .
- This leads to corollaries regarding the vanishing of $F_i(Mn+r) \pmod 2$ based on whether $r$ belongs to a specific set of quadratic residues.
Congruences Modulo 4:
- The paper establishes specific values for $F_0(n)$ and $F_1(n)$ modulo 4 depending on whether $n$ is a square, twice a square, or neither.
- A recursive relation involving the sum-of-divisors function $\sigma(k)$ is derived for $n F_i(n)$ modulo 2.
Congruences Modulo 8:
- Detailed characterizations are provided for $F_0(2n)$ and $F_1(2n)$ modulo 8, distinguishing cases based on whether $n$ is a square of an odd/even integer or twice a square.
- For arguments of the form $4n+1$ , the authors derive a formula involving $\varepsilon(n)$ (which is 1 if $n$ is a pronic number $a(a+1)$ ) and a divisor sum function $D(n)$ involving Jacobi symbols.
- For arguments of the form $4n+3$ , the value of $F_0(4n+3)$ modulo 8 is shown to depend on the parity of the number of representations of $n$ in the form $2P_r + 16P_s$ , where $P_j$ are generalized pentagonal numbers.

Significance
The paper claims significance primarily in resolving open problems left by Andrews and Bachraoui regarding the analytic structure of these specific two-color partition functions. By providing the proofs for the $q$ -series identities, the authors validate the theoretical framework surrounding $F(n)$ and $H(n)$ . Furthermore, the establishment of congruences modulo 2, 4, and 8 offers a deeper arithmetic understanding of the distribution of these partitions. The work serves as a correction to prior literature and provides a foundation for future investigations into divisibility properties of related restricted partition functions, such as those based on the parity of even parts or the total number of parts.