🔢 mathematics

Two-color partitions with evens in one color

This paper investigates integer partitions into two colors (red and blue) where even parts are restricted to the blue color, deriving explicit formulas for specific subsequences and establishing new partition identities based on parity and color constraints.

Original authors: George E. Andrews, Mohamed El Bachraoui

Published 2026-05-25

📖 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 massive, infinite toy factory. In this factory, you build "numbers" out of smaller blocks. For example, the number 4 can be built using four 1-blocks, or a 3-block and a 1-block, or two 2-blocks. In mathematics, this is called an integer partition.

Now, imagine this factory has a special rule: every block comes in two colors, Red and Blue. This turns the factory into a "two-color" toy shop.

The Big Rule: The Blue-Only Even Blocks

The main story of this paper revolves around a strict safety regulation in the factory: Even-numbered blocks (2, 4, 6, etc.) are only allowed to be Blue.

You can have a Blue 2, a Blue 4, or a Blue 6.
You cannot have a Red 2 or a Red 4.
Odd-numbered blocks (1, 3, 5) can be either Red or Blue.

The authors, George Andrews and Mohamed El Bachraoui, are counting how many different ways you can build these numbers following this rule. They call this total count F(n).

The Mystery of the "Even/Odd" Counts

The authors didn't just stop at counting the total number of ways. They started asking more specific questions about the mix of colors in the toys:

The Red Odd Question: If you look at a specific toy (say, the number 5), how many ways can you build it if the number of Red Odd blocks is an even number (0, 2, 4...)? Let's call this count F0. How many ways if that number is odd? Let's call this F1.
The Blue Even Question: How many ways can you build the toy if the total number of Even blocks (which must be Blue) is even? Let's call this F2. How many if it's odd? Let's call this F3.

The "Magic Formulas" (Theorems)

The paper's main achievement is finding "magic formulas" (mathematical equations) that predict exactly how many ways you can build these toys for any number $n$ .

Theorem 1 & 2: They found a way to calculate F0 and F1.
- The Analogy: Think of this as finding a secret recipe. Instead of manually counting every single way to build a giant tower of blocks, the formula tells you the answer instantly.
- The Surprise: These formulas are surprisingly complex, involving patterns related to the number 16. The authors also discovered that these specific counts are actually equal to the number of ways to build toys with a different, slightly more complicated set of rules (involving "overlined" blocks, which are like blocks with a little hat on them).
Theorem 3 & 4: They found formulas for F2 and F3.
- The Analogy: They connected the "Even/Odd" counts of their blue-even-blocks factory to a concept called the "Minimal Excludant" (mex).
- What is Mex? Imagine you are building a tower. You look at the numbers 2, 6, 10, 14... (numbers that are 2 more than a multiple of 4). The "Mex" is the smallest number in that list that is missing from your tower.
- The Discovery: The authors proved that the number of ways to build a tower where the "missing number" follows a specific pattern is exactly the same as the number of ways to build a tower where the total count of even blocks is even (or odd). It's like saying, "The number of ways to have an even number of blue bricks is exactly the same as the number of ways to be missing a specific type of brick."

The Connection to "Overpartitions"

The paper also reveals a deep link between their two-color factory and a different type of factory called the "Overpartition" factory.

In an overpartition, you can put a "hat" (an overline) on the first time a number appears.
The authors showed that their total count F(n) is actually the same as the number of overpartitions of $n$ .
They also broke down their counts (F0 and F1) into simple combinations of overpartition numbers. It's like realizing that your complex two-color toy factory is actually just a disguised version of the simpler "hat" factory.

The "No-Repeat" Subset

They also looked at a stricter version of the factory (H(n)) where you cannot use the same block size twice in the same color.

They found that if you count these "no-repeat" toys based on whether the total number of blocks is even or odd, the answers depend on whether the number you are building is a perfect square (like 1, 4, 9, 16).
If the number is a perfect square, the counts shift slightly; if not, they split perfectly in half.

The Open Questions

Finally, the authors admit that while they proved these formulas using complex algebra (manipulating infinite series), they haven't found a visual, step-by-step "bijection" (a direct one-to-one matching) to explain why these things are equal.

The Challenge: They are asking other mathematicians to find a way to physically pair up every "Red-Odd-Even" toy with a "Missing-Number" toy without using the magic formulas, just by looking at the blocks.

Summary

In short, this paper is about:

Counting ways to build numbers using Red and Blue blocks, where even numbers must be Blue.
Discovering that counting these specific arrangements leads to beautiful, complex mathematical formulas.
Realizing that these counts are secretly the same as counting "hatted" numbers (overpartitions) or counting based on which numbers are missing from the set.
Challenging the math world to find a visual explanation for these surprising connections.

Technical Summary: Two-Color Partitions with Evens in One Color

Problem Statement
This paper investigates sequences of integer partitions in two colors (red and blue) subject to specific parity and color constraints. The primary object of study is the set $F(n)$ , defined as the collection of two-color partitions of an integer $n$ where even parts are restricted to appear only in the blue color. The authors analyze subsequences of $F(n)$ based on the parity of the count of specific types of summands:

$F_0(n)$ and $F_1(n)$ : The number of partitions in $F(n)$ where the number of odd red parts is even or odd, respectively.
$F_2(n)$ and $F_3(n)$ : The number of partitions in $F(n)$ where the total number of even parts is even or odd, respectively.

Additionally, the paper introduces a subset $H(n)$ where parts of the same color do not repeat, and analyzes similar parity-based subsequences ( $H_0$ through $H_3$ ). The study also connects these sequences to the theory of overpartitions and minimal excludants ($mex$).

Methodology
The authors employ analytic methods rooted in the theory of $q$ -series. The methodology relies on:

Generating Functions: Constructing generating functions for the sequences $F(n)$ , $F_i(n)$ , and $H_i(n)$ using standard $q$ -product notation, such as $(a; q)_\infty = \prod_{j=0}^\infty (1-aq^j)$ .
Algebraic Manipulation: Utilizing identities such as Euler's formula and the Jacobi Triple Product Identity to transform and simplify these generating functions.
Series Expansion: Expressing the generating functions as infinite sums to identify coefficients and establish combinatorial equalities.
Comparative Analysis: Comparing the derived generating functions with known generating functions for overpartitions ( $p(n)$ ) and overpartitions into odd parts ($po(n)$) to deduce explicit formulas.

Key Contributions and Results

Generating Functions and Combinatorial Interpretations (Theorems 1 & 2):
The authors establish closed-form generating functions for $F_0(n)$ and $F_1(n)$ .
- $\sum F_0(n)q^n$ is expressed as a product involving terms modulo 16. This leads to Corollary 1, which interprets $F_0(n)$ as the number of partitions of $n$ with optional overlines, where odd parts and parts $\equiv 0 \pmod{16}$ are restricted to blue, while parts $\equiv 6, 10 \pmod{16}$ may appear in both colors with specific overlining rules.
- $\sum F_1(n)q^n$ is similarly derived, leading to Corollary 2, which provides a combinatorial interpretation for $F_1(n)$ involving partitions of $n-1$ with parts $\equiv 2, 14 \pmod{16}$ .
Connections to Overpartitions (Theorem 5):
The paper derives explicit formulas relating the two-color sequences to standard overpartitions $p(n)$ :
- $F_0(n) = p(n) + \frac{p(n/2)}{2}$
- $F_1(n) = p(n) - \frac{p(n/2)}{2}$
  These identities demonstrate that the sequences $F_0$ and $F_1$ are linear combinations of overpartition counts.
Minimal Excludant Identities (Theorems 3 & 4):
The authors extend the concept of the minimal excludant ($mex$) to two-color partitions. They prove that:
- $F_2(n) = p_{4,2}(n, \text{blue})$
- $F_3(n) = \overline{p}_{4,2}(n, \text{blue})$
  Here, $p_{4,2}(n, \text{blue})$ counts partitions where the smallest missing part congruent to $2 \pmod 4$ satisfies specific modular conditions relative to the blue parts.
Results for Non-Repeating Parts (Theorem 6):
For the subset $H(n)$ (where same-color parts do not repeat), the authors provide formulas for $H_0, H_1, H_2, H_3$ in terms of $po(n)$ (overpartitions into odd parts). These formulas depend on whether $n$ is a perfect square, involving terms like $\frac{po(n)}{2} \pm 1$ or $\frac{po(n)}{2} \pm (-1)^n$ .
New $q$ -Series Identities:
In the concluding section, the authors derive new $q$ -series identities by analyzing the generating functions based on the greatest and smallest parts of the partitions (Equations 5.3 and 5.6).

Significance and Claims
The paper claims to establish new partition identities and generating functions for two-color partitions with specific parity constraints. It highlights the natural emergence of these sequences in the theory of integer partitions, specifically linking them to overpartitions.

The authors explicitly state that their proofs for the main theorems and corollaries are analytic, relying on $q$ -series manipulations. They identify a gap in the current work by posing Open Problem 1: to provide bijective proofs for Corollaries 1–2 and Theorems 3–4. Furthermore, they present new $q$ -series identities in Section 5 and pose Open Problem 2 regarding the derivation of analytic proofs for these specific identities. The work is presented as a contribution to the combinatorial theory of partitions and $q$ -series, without claiming immediate applications to other fields.