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Parity of parts and excludant statistics in partitions

This paper investigates restricted excludant statistics in partitions with distinct parts of the same parity, establishing connections between their generating functions and specific quantum modular forms via $q$ -series transformations, and deriving their asymptotic behavior using Tauberian methods.

Original authors: Gargi Mukherjee

Published 2026-03-17

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Gargi Mukherjee

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 librarian in a magical library where every book represents a number. But instead of just stacking them, you have a special rule: you can only arrange books in piles where the odd-numbered books (1, 3, 5, 7...) can appear only once in a pile. You can have as many even-numbered books (2, 4, 6, 8...) as you like, but the odd ones must be unique.

This paper is about a game played with these specific piles of books (mathematical "partitions"). The author, Gargi Mukherjee, introduces a new way to score these piles based on a concept called the "Excludant."

The Game: Finding the Missing Piece

Imagine you have a pile of books. You look at the numbers 1, 2, 3, 4, 5... in order.

If your pile has a 1, you check 2.
If it has a 2, you check 3.
You keep going until you find the first number that is missing from your pile.

This missing number is the Excludant.

If your pile is {6, 4, 2}, the numbers 1, 3, 5 are missing. The first missing one is 1.
If your pile is {5, 1}, the number 2 is missing. The excludant is 2.

The author asks two big questions about these missing numbers:

Is the missing number odd or even? (Parity)
How big is the missing number? (Summing them up)

The "Odd-Only" Rule (The Main Character)

The paper focuses heavily on a specific type of pile: Distinct Odd Parts. This means you can have a 1, a 3, and a 5, but you cannot have two 3s. You can have as many even numbers as you want, but the odd ones are the VIPs who must be unique.

1. The "Odd vs. Even" Missing Number Count

The author counts how many times the missing number is odd versus how many times it is even.

The Analogy: Imagine a dance floor. Some dancers are missing a partner with an "Odd" number on their shirt, and others are missing a partner with an "Even" number. The author creates a formula (a "generating function") that acts like a magical calculator. If you type in the size of the pile, it tells you exactly how many dancers are missing an odd partner versus an even partner.
The Discovery: The author found that these counts aren't random. They are deeply connected to ancient mathematical formulas discovered by the legendary mathematician Ramanujan. It's like finding that the pattern of raindrops on your window follows the same secret code as the stars in the sky.

2. The "Sum of Missing Numbers"

Instead of just counting how many times a number is missing, the author adds up the values of all the missing numbers for every possible pile.

The Analogy: Imagine every missing number is a coin in a jar. The author calculates the total weight of the jar for piles of size $n$ .
The Discovery:
- When the missing number is the smallest odd number missing, the total weight follows a very specific, elegant curve related to Ramanujan's work.
- When the missing number is the smallest even number missing, the total weight is surprisingly simple: it is exactly twice the number of ways to make a pile using only distinct even numbers. It's like finding a shortcut: "If you want to know the total weight of missing even numbers, just count the even-only piles and double it!"

3. The "Biggest Missing Number"

The author also looked at the largest number that is missing but is smaller than the biggest number in the pile.

The Analogy: Imagine you have a shelf of books up to size 10. If you are missing a 9, but you have a 10, the "Max Excludant" is 9. If you have 10, 8, and 6, the biggest missing number below 10 is 9.
The Discovery: This is a much harder puzzle. The formulas here are more complex, involving "Mock Theta Functions" (a fancy type of mathematical function Ramanujan was obsessed with). It's like solving a Rubik's cube where the colors keep shifting, but the author managed to write down the instructions to solve it.

The "Growth" of the Game (Asymptotics)

The paper also asks: "As the piles get huge, what happens?"

The Analogy: If you keep making bigger and bigger piles of books, does the number of "odd missing" piles grow faster than the "even missing" piles?
The Discovery: Yes! The author proved that as the numbers get very large, the number of piles with an odd missing number grows at a specific, predictable rate (like a rocket taking off). They used a mathematical tool called "Tauberian methods" (think of it as a telescope) to look at the distant future of these numbers and predict their behavior with high precision.

The "Staircase" Trick (Combinatorics)

One of the coolest parts of the paper is a combinatorial proof (a visual, logic-based proof) for one of the identities.

The Analogy: The author built a bridge between two different worlds.
- World A: Piles with distinct odd numbers.
- World B: Piles with distinct even numbers, but colored in two different colors (like red and blue).
The author showed a "staircase" method to take a pile from World A, cut it, rotate it, and reassemble it into a pile in World B. It's like showing that a pile of red bricks and a pile of blue bricks are actually made of the exact same amount of clay, just arranged differently. This proves that the two seemingly different mathematical quantities are actually equal.

Why Does This Matter?

You might ask, "Who cares about missing numbers in piles?"

The Big Picture: Mathematics is full of hidden patterns. This paper connects simple counting games (partitions) to the deepest, most complex structures in math (Quantum Modular Forms and Ramanujan's functions).
The Legacy: Ramanujan left behind many unsolved mysteries. This paper solves a few small pieces of those puzzles and shows how modern tools can unlock ancient secrets.
The Future: The author ends by asking new questions, inviting other mathematicians to try and find simpler ways to prove these results or to find even deeper connections.

In a nutshell: This paper is a detective story. The detective (the author) investigates a specific type of number pile, finds a hidden "missing number" pattern, realizes it's connected to a famous historical code (Ramanujan's), proves the connection using both magic formulas and visual puzzles, and predicts how the pattern behaves in the infinite future.

1. Problem Statement

The paper investigates restricted excludant statistics within the theory of integer partitions, specifically focusing on the parity (odd/even nature) of these statistics. The study is confined to two specific classes of partitions:

Partitions into distinct odd parts (denoted as $Pod(n)$): Partitions where odd parts appear at most once, but even parts can appear with any multiplicity.
Partitions into distinct even parts (denoted as $Ped(n)$): Partitions where even parts appear at most once, but odd parts can appear with any multiplicity.

The author defines and analyzes four primary statistics for these sets:

Minimal Excludant ($mex$): The smallest positive integer not present in the partition.
Maximal Excludant ($max$): The largest non-negative integer smaller than the largest part of the partition that is not present in it.
Odd/Even Variants: The paper further refines these by considering the smallest/largest odd or even integers missing from the partition (e.g., $moex$, $meex$, $moax$, $meax$).

The core problem is to determine the generating functions for the counts of partitions where these excludants have specific parities, and to derive the asymptotic behavior of these sequences.

2. Methodology

The author employs a combination of analytic number theory and combinatorial techniques:

$q$ -Series Transformations: The primary tool for deriving generating functions. The paper utilizes:
- The $q$ -Binomial theorem (Cauchy).
- Gauss's identities and Heine's transformation formula.
- Andrews' transformation formulas.
- Connections to Ramanujan's mock theta functions (specifically $\nu(q)$ , $\sigma(q)$ , and $\sigma^*(q)$ ) and Nahm-type sums ( $v_2(q)$ ).
Combinatorial Arguments: Used to prove monotonicity and establish bijections.
- Injective Maps: Constructing maps between sets of partitions of $n$ and $n+1$ to prove strict inequalities (monotonicity).
- Bijections: Constructing explicit bijections between partitions with restricted excludants and two-colored partitions to prove identities (e.g., relating $\sigma_{od}^{meex}(n)$ to $2 \cdot ped(n)$ ).
Tauberian Theorems: Specifically Ingham's Tauberian theorem, is used to derive the asymptotic growth rates of the partition statistics from their generating functions. This involves analyzing the behavior of the generating functions as $q \to 1^-$ (or $t \to 0^+$ where $q=e^{-t}$ ).

3. Key Contributions and Results

A. Generating Functions for Parity of Minimal Excludants ($Pod(n)$)

The paper derives closed-form generating functions for the number of partitions in $Pod(n)$ where the minimal excludant is odd ( $ao_{od}(n)$ ) or even ( $ae_{od}(n)$ ).

Result: These generating functions are expressed in terms of $v_2(q)$ (a Nahm-type sum) and infinite products.
Significance: This links the parity of the minimal excludant to known modular and mock modular forms.

B. Asymptotics

Using Tauberian methods, the author establishes the asymptotic behavior as $n \to \infty$ :

$ao_{od}(n) \sim \frac{e^{\pi\sqrt{n}/2}}{4\sqrt{2n}}$
$ae_{od}(n) \sim \frac{e^{\pi\sqrt{n}/2}}{\sqrt{2n}}$
These results show that while both sequences grow exponentially, the count of partitions with an even minimal excludant is asymptotically four times larger than those with an odd minimal excludant.

C. Sum of Excludants ( $\sigma$ statistics)

The paper studies the sum of the minimal excludants over all partitions in $Pod(n)$, separated by the parity of the excludant ( $\sigma_{od}^{moex}$ and $\sigma_{od}^{meex}$ ).

Key Identity: The sum of the smallest even missing parts ( $\sigma_{od}^{meex}(n)$ ) is exactly twice the number of partitions into distinct even parts:
$\sigma_{od}^{meex}(n) = 2 \cdot ped(n)$
This is proven both analytically via $q$ -series and combinatorially via a bijection involving "staircase profiles" and two-colored partitions.
Connection to Mock Theta Functions: The generating function for the sum of smallest odd missing parts ( $\sigma_{od}^{moex}(n)$ ) is related to Ramanujan's companion function $\sigma^*(q)$ .

D. Maximal Excludants

The paper extends the analysis to the maximal excludant ($max$) for $Pod(n)$.

Generating functions for the parity of $max(\lambda)$ are derived, involving Ramanujan's third-order mock theta function $\nu(q)$ .
Similar sums ( $\sigma_{od}^{moax}$ and $\sigma_{od}^{meax}$ ) are analyzed, with generating functions expressed via complex $q$ -hypergeometric series.

E. Partitions into Distinct Even Parts ($Ped(n)$)

The study is mirrored for $Ped(n)$:

Generating functions for the parity of $mex$ are derived.
Sums of minimal excludants ( $\sigma_{ed}^{moex}$ and $\sigma_{ed}^{meex}$ ) are related to $\sigma(q)$ (Ramanujan's mock theta function) and modular forms.

F. Monotonicity

The paper proves that the sequences counting partitions with specific excludant parities are strictly increasing for sufficiently large $n$ .

$ao_{od}(n+1) > ao_{od}(n)$ for $n \ge 1$ .
$ae_{od}(n+1) > ae_{od}(n)$ for $n \ge 7$ .
Similar monotonicity is proven for the sum statistics $\sigma_{od}^{moex}(n)$ .

4. Significance

Bridging Fields: The work successfully bridges combinatorial partition theory with the theory of modular forms and mock theta functions, demonstrating that simple parity conditions on excludants yield deep connections to Ramanujan's work.
New Identities: It provides new combinatorial identities, such as the exact relation between the sum of even minimal excludants and the count of distinct even partitions.
Asymptotic Precision: It offers precise asymptotic formulas for these restricted statistics, which are often difficult to obtain without Tauberian methods.
Open Problems: The paper concludes by posing significant open questions, including:
- The modularity of specific $q$ -hypergeometric series found in the generating functions.
- The existence of Rademacher-type exact formulas for these sequences.
- Combinatorial proofs for the monotonicity of these sequences (currently proven via injective maps).
- The relationship between the generating functions of $\sigma_{od}^{moex}$ and $\sigma_{ed}^{meex}$ in the context of roots of unity.

In summary, this paper provides a comprehensive analysis of how the parity of excludant statistics influences the structure of restricted partitions, utilizing advanced $q$ -series machinery to uncover deep connections to classical modular forms and establishing rigorous asymptotic and combinatorial results.