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Relations for partitions with distinct even parts except the largest part which is even

This paper establishes new $q$ -series identities and congruences connecting 4-regular partitions with partitions featuring distinct even parts where the largest part is even, while also introducing and analyzing three related partition functions.

Original authors: Gaurab Bardhan, Nipen Saikia

Published 2026-02-18

📖 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 a master chef in a bustling kitchen called Partition Land. In this kitchen, the main ingredient is the number $n$ (like a large cake you need to slice). Your job is to cut this cake into smaller pieces (parts) that add up to the original size.

Usually, there are rules about how you can cut the cake. This paper is about two specific, quirky sets of rules that the authors, Gaurab and Nipen, have been studying. They are trying to figure out how many different ways you can cut the cake under these rules, and how these two different cutting styles are secretly related to each other.

Here is the breakdown of their discovery in simple terms:

1. The Two Types of Cakes

The paper focuses on two special "flavors" of partitioning:

Flavor A: The "No-Divisible-by-4" Cake (4-Regular Partitions)
Imagine you have a rule: "You cannot use any slice size that is a multiple of 4." So, you can't have slices of size 4, 8, 12, etc. You can have 1, 2, 3, 5, 6, 7, 9...
- The Math: This is called a 4-regular partition. The authors use this as a "control group" or a baseline to measure everything else against.
Flavor B: The "Even-Parts-Only-Once" Cake (Distinct Even Parts)
Imagine a different rule: "You can have as many odd slices as you want, but if you use an even slice (2, 4, 6...), you can only use that specific size once."
- Example: You can have a 4 and a 2, but you can't have two 4s.
- The Twist: The authors realized that the number of ways to make "Flavor A" cakes is exactly the same as the number of ways to make "Flavor B" cakes. It's like discovering that a secret code exists between two completely different languages.

2. The New Characters: The "Largest Part" Rules

The authors didn't stop there. They invented three new, very specific characters (partition functions) to play with. Think of these as special chefs with very picky habits regarding the largest slice of the cake.

Chef DEe: This chef says, "I want all even slices to be unique (no repeats), EXCEPT for the biggest slice. The biggest slice can be even, and it can be repeated as much as I want!"
- Analogy: Imagine a pizza where every topping appears only once, but the "King Topping" (the biggest one) can be piled on top of itself.
Chef DEe1: This chef is stricter. "The biggest slice must be even, and it must appear exactly once."
Chef DEe≥k: This chef says, "The biggest slice must be even, and it must appear at least $k$ times."

3. The Big Discovery: The Connection

The main point of the paper is that the authors found a magic bridge connecting these picky chefs (DEe, DEe1, etc.) to the "No-Divisible-by-4" cake (4-regular partitions).

They proved mathematical formulas (identities) that show:

"If you count how many ways Chef DEe can cut a cake of size $n$ , it is directly related to counting the ways to cut a 'No-Divisible-by-4' cake of a slightly different size."

It's like saying: "If you know how many ways you can arrange a deck of cards without any 4s, you can instantly calculate how many ways Chef DEe can arrange a cake with a repeated biggest slice."

4. The "Modulo" Mystery (The Remainder Game)

The second half of the paper is like a game of remainder.

Imagine you have a huge pile of cakes. Instead of counting the exact number of ways to cut them (which can be millions), the authors ask: "If we divide the number of ways by 2, 4, or 8, what is the remainder?"

They found patterns. For example, they proved that for certain cake sizes, the number of ways Chef DEe can cut the cake is always an even number (remainder 0 when divided by 2).
They used a tool called Triangular Numbers (1, 3, 6, 10, 15...) as a map to predict these remainders. It's like saying, "If the cake size is 10, the answer is even. If it's 11, the answer is odd."

Why Does This Matter?

You might ask, "Who cares about cutting imaginary cakes?"

In the world of mathematics, these "cakes" represent deep structures in numbers.

Simplifying Complexity: By proving these connections, the authors turned a hard problem (counting complex partitions) into an easier one (counting 4-regular partitions).
Predicting Patterns: The "remainder" rules (congruences) help mathematicians predict properties of numbers without doing the heavy lifting of counting every single possibility.
New Tools: They created new formulas (q-series identities) that other mathematicians can use as tools to solve even harder problems in the future.

Summary in One Sentence

The authors discovered that the number of ways to cut a cake with unique even slices (with a special rule for the biggest slice) is mathematically linked to the number of ways to cut a cake without any slices divisible by 4, and they figured out the exact "remainder" patterns for these counts.

1. Problem Statement and Context

The paper addresses problems in the theory of integer partitions, specifically focusing on the arithmetic properties and generating functions of specific partition classes. The work builds upon recent research by Andrews and Bachraoui (2025), who established relations between:

$ped(n)$: Partitions of $n$ with distinct even parts.
$reg_4(n)$ : 4-regular partitions of $n$ (partitions where no part is divisible by 4).
$DE_1(n), DE_2(n), DE_3(n)$ : Specific variations of partitions with distinct even parts where the largest part is odd, subject to constraints on the multiplicity of the largest part.

The authors aim to:

Generalize the partition functions $DE_2(n)$ and $DE_3(n)$ to arbitrary multiplicities ( $k$ ).
Define three new partition functions where the largest part is even (while all other even parts remain distinct), a condition not previously explored in this specific context.
Derive new $q$ -series identities connecting these new functions to 4-regular partitions.
Establish congruence properties (modulo 2, 4, and 8) for these new functions.

2. Methodology

The authors employ standard techniques from the theory of $q$ -series and partition identities:

Generating Functions: The core methodology involves defining generating functions for the new partition classes using infinite products and sums involving $q$ -Pochhammer symbols $(a; q)_n$ .
$q$ -Series Identities: The proofs rely heavily on classical identities, including:
- The $q$ -Binomial Theorem.
- The Andrews-Subbarao-Vidyasagar summation formula.
- Lebesgue's identity.
- Jacobi's Triple Product Identity.
Algebraic Manipulation: The authors perform complex algebraic manipulations on infinite series to transform generating functions of the new partition classes into forms involving the generating function of 4-regular partitions, $\sum reg_4(n)q^n = \frac{(q^4; q^4)_\infty}{(q; q)_\infty}$ .
Coefficient Extraction: To prove recurrence relations and congruences, the authors equate coefficients of $q^n$ on both sides of derived identities.
Modular Arithmetic: For congruence results, the authors utilize the Binomial Theorem modulo powers of 2 and properties of Ramanujan theta functions to simplify generating functions modulo 4 and 8.

3. Key Definitions and Contributions

The paper introduces several new partition functions and generalizations:

A. Generalizations of Existing Functions

$DE_{\ge k}(n)$ : Partitions of $n$ with distinct even parts, where the largest part is odd and appears at least $k$ times.
$DE_k(n)$ : Partitions of $n$ $n$ with distinct even parts, where the largest part is odd and appears exactly $k$ $k$ times.
- Contribution: The authors derive general $q$ -series identities (Theorems 1 and 2) for these functions, linking them to 4-regular partitions.

B. New Partition Functions (Largest Part is Even)

The authors define three new functions where the largest part is even, while all other even parts are distinct:

$DE_e(n)$ : Partitions where no even part is repeated, except the largest part which is even and unrestricted.
$DE_{e1}(n)$ : Partitions where no even part is repeated, and the largest part (even) appears exactly once.
$DE_{e\ge k}(n)$ : Partitions where no even part is repeated, except the largest part (even) which appears at least $k$ times.

4. Main Results

A. New $q$ -Series Identities

Theorem 1 & 2: Provide closed-form $q$ -series representations for the generating functions of $DE_{\ge k}(n)$ and $DE_k(n)$ , expressing them in terms of infinite products involving $(q^4; q^4)_\infty$ and $(q; q)_\infty$ .
Theorem 3 & 4: Establish linear recurrence relations connecting $DE_2(n)$ and $DE_{\ge 3}(n)$ to $reg_4(n)$ . For example, Theorem 3 shows a linear combination of shifted $DE_2$ terms equals a linear combination of shifted $reg_4$ terms.

B. Relations to 4-Regular Partitions

Theorem 5: Establishes a direct summation formula for $DE_e(n)$ :
$DE_e(n) = \sum_{i=0}^{\lfloor \frac{n-1}{3} \rfloor} (-1)^i reg_4(n - 1 - 3i)$
Theorem 6: Establishes a similar alternating sum for $DE_{e1}(n)$ :
$DE_{e1}(n) = \sum_{i=0}^{n-2} (-1)^i reg_4(n - 1 - i)$
Theorem 7: Relates $DE_{e\ge 2}(n-2) + DE_{\ge 3}(n)$ to 4-regular partitions with parts strictly greater than 1 ( $reg_{4>1}$ ).

C. Congruence Properties

The paper proves significant congruence properties modulo 2, 4, and 8:

Modulo 2 (Theorems 8 & 9): The authors derive relations involving triangular numbers ( $T_j = j(j+1)/2$ ) and squares ( $2j^2$ ). For instance, they show that sums of $DE_e$ values shifted by triangular numbers are congruent modulo 2 under specific conditions.
Modulo 4 and 8 (Theorems 11 & 12): By analyzing the generating function modulo 4 and 8 using binomial expansions, the authors derive congruences involving sums of $DE_e$ and $DE_{e1}$ weighted by coefficients related to squares. These results generalize previous findings on partition congruences.

5. Significance and Impact

Extension of Partition Theory: The paper successfully extends the study of partitions with distinct even parts to cases where the largest part is even, a constraint that significantly alters the combinatorial structure and generating functions.
Unification of Partition Classes: By linking these new, complex partition functions to the well-studied 4-regular partitions ( $reg_4$ ), the authors provide a bridge that allows arithmetic properties of $reg_4$ to be transferred to these new classes.
New Congruences: The derivation of congruences modulo 4 and 8 for these specific partition types adds to the body of knowledge regarding the arithmetic behavior of partition functions, which is a central theme in modern number theory (following the legacy of Ramanujan).
Generalization Framework: The definitions of $DE_{\ge k}$ and $DE_k$ provide a framework for future research into partitions with specific multiplicity constraints on the largest part.

In conclusion, this paper provides a rigorous analytical framework for a new class of partitions, deriving explicit generating functions, recurrence relations, and deep arithmetic congruences, thereby enriching the landscape of partition identities and $q$ -series.