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An Exploration of Crank Generating functions for $t$ -core partitions

This paper introduces a family of crank generating functions designed to explain specific partition congruences for $t$ -core partitions, extending the combinatorial understanding of Ramanujan's congruences originally addressed by Dyson's crank conjecture.

Original authors: Samuel Wilson

Published 2026-03-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Samuel Wilson

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 have a giant, infinite bag of LEGO bricks. You want to build towers using these bricks, but you have a rule: you can only use a specific number of bricks to build each tower. The question mathematicians have asked for over a century is: "How many different ways can I build a tower of exactly $n$ bricks?"

This number is called a partition. For example, if you have 4 bricks, you can build them in 5 different ways (4, 3+1, 2+2, 2+1+1, 1+1+1+1).

The Mystery of the "Magic Numbers"

In the early 1900s, a genius named Ramanujan noticed something weird. He found that if you build towers with a specific number of bricks (like 5, 7, or 11), the number of ways to build them follows a secret pattern. Specifically, if you try to build a tower with $5n + 4$ bricks, the number of ways to do it is always divisible by 5. It's like the universe is whispering a secret code: "Every time you hit this number, the answer is a multiple of 5."

For a long time, mathematicians knew the code existed, but they didn't know why. It was like knowing a lock always opens at 3:00 PM, but not knowing what key turns it.

The "Crank": The Magic Key

In the 1940s, a mathematician named Freeman Dyson invented a tool called the Rank to explain the first two secrets. But he still couldn't crack the third one. So, he guessed there must be another, more powerful tool hidden inside the towers. He called this hypothetical tool the Crank.

Decades later, in 1988, two other mathematicians found the Crank. It's a way of looking at a tower and assigning it a "score" based on its shape. When you sort all the towers of a certain size by their Crank scores, they split up perfectly into equal groups. This explains why the numbers are divisible by 5, 7, or 11. It's like sorting a deck of cards: if you deal them out into 5 piles, and every pile has the exact same number of cards, you know the total must be divisible by 5.

The New Adventure: "Core" Towers

The author of this paper, Samuel Wilson, decided to take this idea of the Crank and apply it to a special, stricter type of tower called a $t$ -core partition.

Think of a standard tower as a stack of blocks where you can have any size blocks. A $t$ -core tower is a tower where you are forbidden from having any "hooks" of size $t$ . It's like a tower that has been "pruned" or "cored" to remove specific shapes. These are harder to study, and for a long time, no one knew if a "Crank" existed for them.

Recently, other mathematicians found that these special towers also follow secret divisibility rules (congruences), but they couldn't explain why using a Crank.

What This Paper Does

Samuel Wilson's paper is like finding the missing keys for a new set of locks.

The Discovery: He built a family of new "Crank" formulas specifically for these $t$ -core towers.
The Proof: He showed that for specific sizes of these towers (like 5-core, 7-core, 11-core, etc.), if you sort them by his new Crank score, they split up perfectly into equal groups.
The Result: This explains why the number of ways to build these special towers is divisible by 3 (in the specific cases he studied).

The Analogy of the "Half-Empty Glass"

Wilson admits his work isn't the final answer to everything. He found keys for some of the locks, but not all of them.

The Good News: He found the Crank for 5-core, 7-core, 11-core, 17-core, and 19-core towers.
The Missing Piece: For some numbers (like 13 or 23), the "Crank" he found only explains half of the secret patterns. It's like having a key that opens the front door but not the back door.
The Mystery: He also notes that for very large numbers (greater than 23), we don't even know if these secret patterns exist yet.

Why Should You Care?

This might sound like abstract math, but it's actually a story about finding order in chaos.

The Problem: Nature (and numbers) often look random.
The Solution: Mathematicians are like detectives looking for the hidden "Crank" that proves the randomness is actually a structured, predictable pattern.
The Impact: Every time we find a new Crank, we understand the deep structure of numbers a little better. It helps us understand how complex systems organize themselves, which has applications in physics, computer science, and cryptography.

In short: Samuel Wilson found a new way to sort special mathematical "towers" to prove why they follow secret counting rules. He didn't solve every mystery, but he handed us a powerful new tool to keep digging for the answers.

1. Problem Statement

The paper addresses a fundamental problem in the theory of integer partitions: providing combinatorial explanations for Ramanujan-type congruences.

Background: Ramanujan famously proved congruences for the partition function $p(n)$ modulo 5, 7, and 11. Freeman Dyson introduced the "rank" to explain the mod 5 and 7 cases, and conjectured the existence of a "crank" statistic to explain all three. Andrews and Garvan later defined the crank and provided its generating function.
The Gap: While crank generating functions have been established for standard partitions and $k$ -color partitions, there was a lack of such functions for $t$ -core partitions.
Specific Objective: The author aims to construct a family of crank generating functions for $t$ -core partitions, denoted $p^{(t)}(n)$ , that explain specific congruences of the form $p^{(t)}(\alpha n + \beta) \equiv 0 \pmod{\ell}$ . Specifically, the paper focuses on congruences modulo 3 for $t$ -cores where $t \in \{5, 7, 11, 17, 19\}$ .

2. Methodology

The paper employs a combination of analytic number theory, modular forms, and algorithmic verification.

Generating Functions: The author defines specific generating functions $C^{(t)}(z, \tau)$ involving infinite products of the form:
$\prod_{n=1}^{\infty} \frac{(1-q^n)^A \prod (1-q^{tn})^B \prod (1-\zeta^{\pm k}q^{tn})}{\prod (1-\zeta^{\pm 1}q^n)}$
where $\zeta = e^{2\pi i z}$ and $q = e^{2\pi i \tau}$ . The goal is to show that the coefficient of $q^{\alpha n + \beta}$ in this series is divisible by the cyclotomic polynomial $\Phi_3(\zeta)$ .
Radu's Lemma: The core theoretical tool used is Radu's Lemma (specifically Lemma 2.4 from the paper, based on work by Radu and Sellers). This lemma provides a criterion to prove that a $q$ $q$ -series satisfies a family of congruences $a(\alpha n + \beta) = 0$ $a (α n + β) = 0$ .
- The method involves defining a set of parameters $(\alpha, M, N, r, \beta)$ belonging to a set $\Delta^*$ .
- It requires calculating the index of the modular group $[\Gamma : \Gamma_0(N)]$ and finding a complete set of double-coset representatives for $\Gamma_0(N) \backslash \Gamma / \Gamma_{\infty}$ .
- The proof reduces to verifying that a specific bound $\lfloor \nu \rfloor$ is non-negative and that the coefficients vanish up to this bound.
Modular Arithmetic and Verification:
- The author reduces the generating functions modulo $\Phi_3(\zeta)$ to simplify the infinite products.
- The verification of the conditions for Radu's Lemma (calculating $\nu$ , finding representatives, and checking coefficients) was performed using the computer algebra system MAPLE.
- A distinction is made for $t=5$ , where $N/2$ is not square-free, requiring manual calculation of double-coset representatives, whereas other cases follow a more standard pattern.

3. Key Contributions

The paper makes several specific contributions to the literature on partition congruences:

Definition of New Crank Generating Functions: The author explicitly constructs generating functions $C^{(t)}(z, \tau)$ for $t \in \{5, 7, 11, 17, 19\}$ .
Explanation of Modulo 3 Congruences: These functions are proven to explain congruences for $p^{(t)}(n)$ $p^{(t)} (n)$ modulo 3. For example:
- For $t=5$ , the function explains $p^{(5)}(15n + \beta) \equiv 0 \pmod 3$ for $\beta \in \{6, 10, 12, 13\}$ .
- For $t=7, 11, 17, 19$ , similar families of congruences are established with specific arithmetic progressions.
Application of Radu's Lemma to t-cores: The paper successfully adapts Radu's Lemma, previously used for standard and colored partitions, to the specific context of $t$ -core partitions, which have a different generating function structure: $\prod (1-q^{tn})^t / (1-q^n)$ .
Identification of Limitations: The author notes that while these functions explain some congruences, they do not explain all known congruences for these $t$ -cores. Specifically, the constructed functions only account for cases where the residue $\beta$ is a quadratic non-residue modulo 3.

4. Results

The main results are formalized in two theorems:

Theorem 1.2 (Case $t=5$ ):
The generating function $C^{(5)}(z, \tau)$ is a crank generating function for 5-core partitions. It proves that $p^{(5)}(15n + \beta) \equiv 0 \pmod 3$ for $\beta \in \{6, 10, 12, 13\}$ .
- Technical Detail: The proof required finding 8 double-coset representatives manually because $N=45$ does not satisfy the square-free condition required for the standard lemma application.
Theorem 1.3 (Cases $t > 5$ ):
The paper provides explicit infinite product formulas for $C^{(7)}$ , $C^{(11)}$ , $C^{(17)}$ , and $C^{(19)}$ .
- $t=7$ : Explains $p^{(7)}(21n + \beta) \equiv 0 \pmod 3$ for $\beta \in \{8, 11, 17\}$ .
- $t=11$ : Explains $p^{(11)}(33n + \beta) \equiv 0 \pmod 3$ for $\beta \in \{11, 20, 26, 29, 32\}$ .
- $t=17$ : Explains $p^{(17)}(51n + \beta) \equiv 0 \pmod 3$ for $\beta \in \{14, 20, 23, 26, 35, 38, 41, 47\}$ .
- $t=19$ : Explains $p^{(19)}(57n + \beta) \equiv 0 \pmod 3$ for $\beta \in \{14, 17, 26, 35, 38, 41, 44, 50, 56\}$ .
Observation on 13 and 23: The paper notes that while congruences exist for 13-core and 23-core partitions (proven by Meher and Jindal), the author has not yet found crank generating functions for them.

5. Significance and Future Directions

Combinatorial Insight: By providing crank generating functions, the paper offers a potential combinatorial statistic (a "crank") that partitions the set of $t$ -core partitions into equally sized sets based on the residue of the statistic modulo 3. This moves the field from purely analytic proofs of congruences to combinatorial understanding.
Methodological Advancement: The successful application of Radu's Lemma to $t$ -cores demonstrates the versatility of this lemma in handling complex modular forms that are not standard eta-quotients.
Open Questions: The author concludes with three significant open questions:
1. Can a single family of crank generating functions explain all congruences for $p^{(t)}(n)$ simultaneously for $t \le 23$ ? (Currently, the functions only cover half the known congruences).
2. Do similar "mod 3" congruence families exist for $t > 23$ ?
3. Can these specific families be extended to an infinite family of $t$ -cores?

In summary, this paper bridges the gap between the analytic proof of $t$ -core congruences and the combinatorial theory of cranks, providing the first explicit generating functions for this class of partitions modulo 3, while highlighting the complexity of covering all known congruences with a single statistic.