🔢 mathematics

Restricted Overpartitions and concave compositions: their modularity and asymptotics

This paper investigates restricted overpartitions and concave compositions, demonstrating that their generating functions exhibit mixed modular structures involving modular forms, mock theta functions, mock Maass theta functions, and false theta functions, while also deriving their asymptotic main terms and analyzing related rank statistics.

Original authors: Koustav Banerjee, Kathrin Bringmann, Atul Dixit

Published 2026-04-03

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Koustav Banerjee, Kathrin Bringmann, Atul Dixit

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 very strict kitchen. Your job is to count how many different ways you can arrange ingredients to make a specific total weight (let's say, a "number" $n$ ). In the world of mathematics, these arrangements are called partitions.

Usually, the rules are simple: you can use any number of ingredients, and the order doesn't matter (just the total count). But in this paper, the authors (Koustav Banerjee, Kathrin Bringmann, and Atul Dixit) are playing with a much more complicated set of rules. They are studying "Restricted Overpartitions" and "Concave Compositions."

Here is a simple breakdown of what they did, using everyday analogies.

1. The Ingredients: What are they counting?

Standard Partitions: Imagine you have a pile of 6 bricks. You can stack them in many ways: a tower of 6, a stack of 3 and 3, a stack of 4 and 2, etc.
Overpartitions: Now, imagine some of your bricks are special. You can put a "highlighter" mark (an overline) on the first time you use a specific size of brick. So, a "3" and a "highlighted 3" are treated as different ingredients. This doubles the possibilities.
Restricted Overpartitions: The authors added a twist. They said, "Okay, you can use these special highlighted bricks, BUT..."
- Rule A: You can't highlight the smallest brick.
- Rule B: If the smallest brick is odd, you can only use it once.
- Rule C: If the smallest brick is even, it can't be highlighted.
- They counted how many ways you can build the number 6 under these strict rules. (The answer is 7 ways).
Concave Compositions: Imagine building a mountain. You start with a small pile, go up to a peak (the "central part"), and then go back down. The numbers must go up, hit a peak, and go down. The authors added rules about which parts of the mountain can be "highlighted" (overlined).

2. The Mystery: The "Magic Formulas"

In math, when you count these things, you usually write down a "generating function." Think of this as a magic recipe card that, if you plug in a number, tells you exactly how many ways you can build that number.

For simple partitions, the recipe card is a smooth, predictable curve. Mathematicians call this a Modular Form. It's like a perfectly round, symmetrical wheel that rolls smoothly.

However, the authors discovered that for their restricted rules, the recipe cards are messy. They aren't just one smooth wheel. They are a Frankenstein's Monster made of four different types of mathematical objects glued together:

Modular Forms: The smooth, predictable wheels.
Mock Theta Functions: These are "almost" smooth wheels. They look like the real thing from a distance, but if you look closely, they wobble. They were a mystery to the famous mathematician Ramanujan until recently.
False Theta Functions: These are tricksters. They look like theta functions (a type of wave), but they have a "twist" (a sign change) that breaks their symmetry. They refuse to behave like normal waves.
Mock Maass Theta Functions: These are even more complex, involving both the "wobbly" nature of mock functions and a specific type of wave equation.

The Big Discovery:
Usually, a counting problem leads to just one type of formula. But the authors found that their specific rules naturally create a mix. It's like trying to bake a cake, but the recipe suddenly requires flour, sand, glitter, and water all at once. It's a "mixed modular structure."

3. The Prediction: Guessing the Future (Asymptotics)

Counting these arrangements for small numbers (like 6 or 10) is easy. But what if you want to know how many ways you can arrange a number as huge as a googol (1 followed by 100 zeros)? You can't count them one by one.

The authors used advanced math tools (like the "Circle Method" and "Tauberian theorems") to find the Asymptotic Main Term.

Analogy: Imagine you are watching a crowd of people enter a stadium. You can't count every single person, but you can look at the speed of the line and the size of the gate to predict, "By 5:00 PM, there will be roughly 50,000 people."
The authors derived a formula that predicts the number of arrangements for very large $n$ with incredible accuracy. They found that even with their weird rules, the numbers grow in a predictable, exponential way (like a population explosion).

4. The Rank: A Scorecard

The authors also looked at a "score" for these arrangements, called the Rank.

Analogy: Imagine a line of people. The "Rank" is the height of the tallest person minus the total number of people in the line.
They asked: "If I look at all the arrangements of the number 100, how many have a Rank of 0? How many have a Rank of 5?"
They found that the formulas for these scores also involve that same "Frankenstein" mix of different mathematical objects.

Why Does This Matter?

You might ask, "Who cares about counting bricks with highlighters?"

The "Mixed" Phenomenon: This paper shows that nature (or at least, the universe of numbers) loves mixing things up. It proves that when you impose specific, logical restrictions on counting problems, you don't just get a simpler answer; you get a complex, beautiful blend of different mathematical worlds.
Solving Old Mysteries: The tools they used help solve old riddles left by Ramanujan. They showed how to "complete" these wobbly functions to make them behave like normal waves, which helps in understanding deep connections in physics and number theory.
New Tools: They developed new ways to predict huge numbers, which is useful in computer science, cryptography, and statistical physics.

Summary

Think of this paper as a group of explorers who found a new island. They discovered that the trees on this island don't just grow leaves; they grow leaves, feathers, scales, and fur all at the same time. They mapped out exactly how these "mixed" trees grow (the asymptotics) and showed that this strange mix isn't a mistake—it's a fundamental rule of how numbers work when you play by specific, interesting rules.

1. Problem Statement

The paper investigates the combinatorial and modular properties of specific families of restricted overpartitions and concave compositions.

Context: While classical partition functions often yield generating functions that are modular forms or mock modular forms (single modular structures), this paper explores restrictions that naturally lead to mixed modular structures.
Specific Objects:
- Restricted Overpartitions: The authors define overpartitions of $n$ with constraints on overlining (e.g., smallest and even parts non-overlined) and parity conditions (e.g., smallest odd part appearing exactly once).
- Concave Compositions: Sums of integers forming a "valley" shape ( $a_1 \ge \dots \ge a_r > c < b_s \le \dots \le b_1$ ) with restrictions on the central part and overlining of odd parts.
Goal: To derive exact generating function identities for these restricted counts, classify their modular nature (identifying components like false theta functions, mock Maass theta functions, and modular forms), and determine their asymptotic behavior as $n \to \infty$ .

2. Methodology

The authors employ a combination of advanced $q$ -series manipulation, the theory of mock modular forms, and analytic number theory techniques.

$q$ -Series Identities: The core of the proofs relies on transforming generating functions using classical and modern $q$ $q$ -series identities. Key tools include:
- Ramanujan's identities (specifically those found in his lost notebook).
- The Rogers–Fine identity.
- Identities by Garvan, Andrews, and Bhoria et al.
- Generalized mock theta functions (e.g., $f(\zeta_1, \zeta_2; q)$ and $\nu(\zeta_1, \zeta_2; q)$ ).
Modular Decomposition: The generating functions are decomposed into sums of distinct modular objects:
- Modular Forms: Standard holomorphic forms (e.g., Dedekind $\eta$ -function related terms).
- Mock Theta Functions: Holomorphic parts of harmonic Maass forms (e.g., Ramanujan's $\phi, \psi, f, \nu$ ).
- Mock Maass Theta Functions: Non-holomorphic eigenfunctions of the hyperbolic Laplacian (completed to modular forms).
- False Theta Functions: Series that resemble theta functions but have sign twists preventing modularity.
Asymptotic Analysis:
- Tauberian Theorems: The authors use a modified version of Ingham's Tauberian Theorem to derive asymptotic formulas for the coefficients $p(n)$ from the behavior of the generating function near the unit circle ( $q \to 1$ ).
- Euler–Maclaurin Summation: Used to analyze the asymptotic behavior of the mock and false theta components as $q = e^{-z} \to 0$ .
- Monotonicity Proofs: To apply Tauberian theorems, the authors first prove that the counting functions are non-negative and weakly increasing.

3. Key Contributions and Results

A. Generating Functions and Modular Decompositions

The paper establishes that the generating functions for these restricted partitions are mixed mock modular forms.

Restricted Overpartitions ($pod(n)$):
- Defined by: Smallest and even parts non-overlined; if smallest is odd, it appears once.
- Result (Theorem 1.1): The generating function $P_{od}(q)$ is expressed as a sum of a mixed mock Maass theta function ( $W_1$ ), a modular form, and a false theta function.
- Refinement: A two-variable generating function $P_{od}(\zeta; q)$ is derived, linking to generalized mock theta functions $f(\zeta_1, \zeta_2; q)$ .
Overpartitions without Repeated Odd Parts ($pev(n)$):
- Defined by: Odd parts non-overlined; specific non-overlining conditions for the smallest even part.
- Result (Theorem 1.3): The generating function $P_{ev}(q)$ is a mixed mock modular form involving Ramanujan's third-order mock theta function $\phi(q)$ .
- Connection: This leads to an identity relating $pev(n)$ to a colored partition function $g(n)$ studied by Andrews and Kumar, yielding the relation $g(n) + pev(n-1) = 2p(n-1)$.
Concave Compositions ($vod(n)$):
- Defined by: Central part even, other parts odd; overlining restrictions on left/right odd parts.
- Result (Theorem 1.14): The rank generating function $V_{od}(\zeta; q)$ is shown to be related to the rank generating functions of standard partitions ( $R$ ) and partitions without repeated odd parts ( $R_2$ ).

B. Asymptotic Formulas

Using the modular decompositions and Tauberian theorems, the authors derive the leading asymptotic terms for the counting functions:

For $pod(n)$ (Corollary 1.2):
$pod(n) \sim \frac{5\pi}{48\sqrt{2}n^{3/2}} e^{\pi\sqrt{5n/6}}$
This growth rate differs from the standard partition function $p(n)$ due to the specific modular weight and the "mixed" nature of the generating function.
For $pev(n)$ (Corollary 1.4):
$pev(n) \sim \frac{e^{\pi\sqrt{2n/3}}}{4\sqrt{3}n}$
Interestingly, this shares the same exponential growth rate as the standard partition function $p(n)$ , despite the complex modular structure.
For $g(n)$ (Corollary 1.5):
$g(n) \sim \frac{e^{\pi\sqrt{2n/3}}}{4\sqrt{3}n}$

C. New Identities

The paper provides new proofs and generalizations of Ramanujan's identities. Notably, Corollary 1.12 recovers a known identity of Ramanujan:
$2\phi(-q) - f(q) = \frac{\Theta^2(-q)}{(q)_\infty}$
using the limit of the two-variable generating function $P_{ev}(\zeta; q)$ as $\zeta \to q^{-2}$ .

4. Significance

Emergence of Mixed Structures: The primary significance is the demonstration that natural combinatorial restrictions on overpartitions do not merely yield single modular objects but simultaneously involve modular forms, mock theta functions, mock Maass theta functions, and false theta functions. This highlights a new class of "mixed modular structures" in partition theory.
Bridging Combinatorics and Analysis: The work connects specific combinatorial definitions (overlining rules) directly to deep analytic properties (modularity and asymptotics), showing how combinatorial constraints dictate the type of modular object required to describe the generating function.
Generalization of Rank Statistics: The study extends the concept of rank statistics to restricted overpartitions, providing a framework for analyzing their distribution and modular properties.
Future Directions: The paper opens several avenues for research, including:
- Determining when other restrictions lead to mixed modular structures.
- Investigating arithmetic congruences for these new counting functions.
- Finding bijective proofs for the derived identities (e.g., the relation between $g(n)$ and $pev(n)$).
- Refining asymptotic expansions using the Circle Method.

In summary, this paper significantly advances the understanding of the modular landscape of partition functions by revealing that restricted overpartitions serve as a rich source of mixed modular phenomena, requiring a synthesis of mock modular forms and false theta functions for their complete description.