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Reciprocals of Subsum Polynomials

This paper introduces the subsum polynomial for integer partitions, investigates the sum of the reciprocals of these polynomials over all partitions of a given integer $n$ , and establishes their arithmetic properties and connections to other combinatorial objects.

Original authors: Cristina Ballantine, George Beck, Brooke Feigon, Kathrin Maurischat

Published 2026-05-12

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Cristina Ballantine, George Beck, Brooke Feigon, Kathrin Maurischat

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 box of Lego bricks. Each brick has a specific size, and you can only use bricks of sizes that are whole numbers (1, 2, 3, etc.).

The Main Characters: Partitions and "Subsum" Polynomials

In this paper, the authors are playing with a concept called an integer partition. Think of a partition as a way to build a tower of a specific total height, $n$ , using your Lego bricks.

If you want a tower of height 4, you could build it with one big 4-brick.
Or a 3-brick and a 1-brick.
Or two 2-bricks.
Or a 2-brick and two 1-bricks.
Or four 1-bricks.

Every unique way you stack these bricks to reach the total height is a "partition."

Now, for every single one of these stacking methods (partitions), the authors create a special mathematical object called a subsum polynomial.

The Recipe: If your tower is made of bricks of sizes $A, B, C$ , the polynomial is $(1 + x^A)(1 + x^B)(1 + x^C)$ .
The Metaphor: Imagine this polynomial is a "menu" of every possible smaller tower you could build using only the bricks you already have in your specific stack. The term $(1 + x^A)$ means "you can either use the $A$ -brick or not use it." When you multiply these together, you get a list of every possible sub-tower you can make from your collection.

The Big Question: The Reciprocal Sum

The authors are curious about what happens if you take the inverse (the reciprocal) of this polynomial for every single possible way to build a tower of height $n$ , and then add them all up.

It's like saying: "For every possible Lego tower of height $n$ , calculate its 'sub-tower menu,' flip that menu upside down, and add all those flipped menus together."

The result of this massive addition is a new, complex fraction made of two polynomials: a numerator (the top part) and a denominator (the bottom part).

What Did They Discover?

The authors spent a lot of time analyzing this resulting fraction. Here are the main things they found, explained simply:

1. The "Greatest Common Divisor" (The Common Thread)
When you add up all these messy fractions, they share a huge common factor in both the top and bottom. The authors found a way to strip this common factor away to get a "simplified" version of the fraction. They call the simplified top part num(n, x) and the bottom part den(n, x).

2. The Shape of the Numbers
They looked at the list of numbers (coefficients) inside these simplified polynomials.

Palindromes: The numbers read the same forwards and backwards, like the word "racecar."
Unimodal: The numbers go up to a peak and then go back down, like a mountain.
The Denominator: The bottom part of the fraction is always "mountain-shaped" (unimodal).
The Numerator: The top part is a palindrome, and they believe (based on computer tests) that it is also mountain-shaped, though they haven't proven it for every single case yet.

3. The "Irreducible" Mystery (Conjecture 1)
The authors have a strong hunch about the top part of the fraction (num(n, x)). They think it is irreducible.

The Metaphor: Imagine the polynomial is a complex machine made of gears. "Irreducible" means you cannot take this machine apart into two smaller, simpler machines that multiply together to make the big one. It is a single, indivisible unit.
They have checked this for small towers (up to height 5) and it holds true. They suspect it's true for all tower heights.

4. Special Values (The "Magic Numbers")
They tested what happens if you plug in specific "magic numbers" (like -1, or imaginary numbers like $i$ ) into these polynomials.

If you plug in -1, the top part of the fraction equals $n!$ (n factorial, which is $1 \times 2 \times 3 \dots \times n$ ). This is a very neat, clean result.
If you plug in other special numbers related to circles (roots of unity), the results follow very specific, predictable patterns involving factorials and powers of 2 or 3.

The "Binary" Side Quest

In Section 4, they looked at a special rule: You can only use bricks that are powers of 2 (1, 2, 4, 8, 16...).

They found that for these specific "binary" towers, the top and bottom parts of the fraction never share any common factors (they are "coprime").
They also found a recursive rule (a recipe to calculate the answer for a big tower based on the answer for a smaller tower) for these binary cases.

The Open Questions (The "To-Do" List)

The paper ends with a few guesses (conjectures) for other types of Lego rules:

Odd Partitions: What if you can only use odd-numbered bricks (1, 3, 5)? They guess the result at -1 is related to factorials.
Ternary Partitions: What if you can only use powers of 3 (1, 3, 9)? They have a similar guess for this case.

Summary

In short, this paper takes a very specific, somewhat abstract way of combining math objects (partitions and polynomials), adds them all up, and discovers that the result has beautiful, symmetrical, and predictable patterns. They proved some of these patterns and made educated guesses about the rest, inviting other mathematicians to solve the remaining puzzles.

Technical Summary: Reciprocals of Subsum Polynomials

Problem Statement
The paper introduces and investigates the "subsum polynomial" associated with an integer partition $\lambda = (\lambda_1, \lambda_2, \dots, \lambda_k)$ . Defined as $sp(\lambda, x) = \prod_{i=1}^k (1 + x^{\lambda_i})$ , this polynomial serves as the generating function for the number of partitions whose parts form a submultiset of $\lambda$ . The primary object of study is the rational function $sr(n, x)$, defined as the sum of the reciprocals of these polynomials over all partitions of an integer $n$ :
$sr(n, x) = \sum_{\lambda \vdash n} \frac{1}{sp(\lambda, x)}$
The authors aim to understand the arithmetic and combinatorial properties of the numerator and denominator of this rational function when expressed in lowest terms.

Methodology
The authors employ a combination of algebraic manipulation, number-theoretic analysis, and combinatorial bijections.

Factorization and GCD Analysis: The rational function $sr(n, x)$ is initially expressed as a ratio of two polynomials, $num^*(n, x)$ and $den^*(n, x)$ . To simplify this, the authors define $G(n, x)$ as the greatest common divisor (GCD) of the numerators of the individual terms in the sum. They analyze the factorization of $G(n, x)$ into terms of the form $(1+x^i)$ , utilizing properties of cyclotomic polynomials and the 2-adic valuation of integers.
Recursive Structures: The paper establishes recursive relationships for the degrees of the resulting polynomials and their values at specific points (roots of unity, integers).
Combinatorial Decomposition: For binary partitions (partitions into powers of 2), the authors use specific bijections (e.g., removing a part of size 1, or halving all parts) to derive recurrence relations for the coefficients of the numerator polynomial.
Evaluation at Roots of Unity: A significant portion of the methodology involves evaluating the numerator polynomial at roots of unity ( $\zeta_{2d}$ ) to determine irreducibility and common factors with the denominator.

Key Contributions and Results

Structure of the Rational Function: The authors define reduced polynomials $num(n, x)$ and $den(n, x)$ by dividing $num^*(n, x)$ and $den^*(n, x)$ by $G(n, x)$ . They prove that $G(n, x)$ , $num(n, x)$, and $den(n, x)$ are all palindromic. Furthermore, $den(n, x)$ and $G(n, x)$ are shown to be unimodal.
Explicit Formulas for Degrees and Values:
- The degree of $den(n, x)$ is calculated as $\sum_{i=1}^n s(i)$ , where $s(i)$ is a sequence related to plane partitions (OEIS A129527).
- The authors derive exact values for $num(n, x)$ and $den(n, x)$ at specific points:
  - $num(n, -1) = n!$
  - $|num(n, \pm i)| = 2^{\lfloor n/2 \rfloor} \lfloor n/2 \rfloor!$
  - Explicit formulas involving factorials and powers of 3 are provided for $|num(n, \zeta_6)|$ .
  - Values at $x=1$ relate to the 2-adic valuation of factorials and the partition function $P(n)$ .
Irreducibility and Coprimality Conjectures: Based on numerical evidence for small $n$ $n$ , the authors propose two central conjectures:
1. $num(n, x)$ is irreducible over the integers for all $n \geq 1$ .
2. $\gcd(num(n, x), den(n, x)) = 1$ .
  The paper notes that the first conjecture implies the second. They provide partial results proving that specific cyclotomic factors do not divide the numerator for certain ranges of $n$ .
Binary Partitions: The study is extended to binary partitions (parts are powers of 2). For this restricted set, the authors prove that the GCD of the numerators is trivial ( $G_B(n, x) = 1$ ). They derive a recurrence relation for the numerator polynomial $num_B(2n, x)$ and establish properties regarding its coefficients, including log-concavity and unimodality conjectures.
Generalizations: The paper briefly considers sums of powers of subsum polynomials, $\sum sp(\lambda, x)^m$ , linking them to partitions with multiple sorts of parts.

Significance and Claims
The paper positions itself as an introduction to the arithmetic properties of sums of reciprocals of partition-generated polynomials. The authors claim to have established foundational arithmetic properties, such as palindromicity and unimodality, and provided exact evaluations at roots of unity.

The significance of the work lies in the identification of deep connections between partition theory, polynomial factorization, and number theory (specifically 2-adic valuations and cyclotomic polynomials). The authors modestly frame their work as laying the groundwork for future exploration, explicitly stating that their irreducibility and coprimality claims are currently supported by numerical evidence and partial proofs rather than general theorems. They conclude by suggesting open questions regarding restricted sets of partitions (odd partitions, ternary partitions) and generalizations involving rising and falling factorials, inviting further research into these combinatorial objects.