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Sum of parts in overpartitions and partitions without repeated odd parts

This paper establishes several Ramanujan-type congruences modulo 5 and 7 for the sums of specific parts in overpartitions and partitions without repeated odd parts, utilizing elementary proofs based on classical theta function identities.

Original authors: Frank Garvan, Rishabh Sarma

Published 2026-06-15

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Frank Garvan, Rishabh Sarma

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 pile of LEGO bricks. In the world of mathematics, a "partition" is simply a way of stacking these bricks to build a tower of a specific height. For example, if your tower needs to be 4 units high, you could build it with four 1-unit blocks, or a 3-unit block and a 1-unit block, or two 2-unit blocks, and so on.

For over a century, mathematicians have been fascinated by a specific rule discovered by the legendary genius Srinivasa Ramanujan. He found that if you count the number of ways to build towers of certain heights (like 5, 10, 15... plus 4), the total count always ends in a zero when divided by 5. It's like a hidden rhythm in the universe of numbers.

This paper, written by Frank Garvan and Rishabh Sarma, takes that idea and applies it to two very specific, slightly more complicated ways of building towers. Instead of just counting how many ways you can build a tower, they are interested in the sum of the parts.

Think of it this way:

Standard Partitions: You just stack bricks.
Overpartitions: Imagine some of your bricks have a little "hat" on them (an overline). A brick with a hat is different from a brick without one.
Partitions without Repeated Odd Parts: You can use as many even-sized bricks as you want, but if you use an odd-sized brick (like a 3 or a 5), you can only use that specific size once. No duplicates allowed for the odd ones.

The authors are asking a new question: "If we add up the sizes of all the bricks in all possible towers of a certain height, does that total sum follow Ramanujan's hidden rhythm?"

The Main Discovery

The authors found that yes, these sums do follow a rhythm, but it depends on which bricks you are counting and which "hat" rules you are using.

They proved that for specific types of towers:

Overpartitions (The Hat Rule): If you look at towers of height $5n+4$ or $5n+2$ , the sum of all the "un-hatted" bricks is perfectly divisible by 5. If you look at towers of height $7n+3$ , that sum is divisible by 7.
Even vs. Odd Bricks: They also separated the bricks by color (even numbers vs. odd numbers). They found that for towers of height $7n+5$ , the sum of the even bricks is divisible by 7. For towers of height $5n+3$ , the sum of the odd bricks is divisible by 5.
The "No Repeated Odd" Rule: When you build towers where odd bricks can't be repeated, they found similar patterns. For example, the sum of even bricks in towers of height $5n$ is divisible by 5.

How They Did It (The "Elementary" Magic)

You might expect a paper with such complex results to use super-advanced, futuristic math. However, the authors proudly state their methods are "elementary."

Think of their proof like a master chef using only basic kitchen tools (knives, pans, and a stove) to create a Michelin-star meal. They didn't need a molecular gastronomy lab; they used classical identities.

In the paper, they use tools called Theta Functions. Imagine these as special recipes or formulas that describe how these numbers behave. The authors took these old, well-known recipes and chopped them up (a process called "dissection") to look at the numbers modulo 5 and modulo 7.

They essentially showed that when you arrange these mathematical "recipes" in a specific way, the terms that would mess up the rhythm (the remainders) cancel each other out perfectly, leaving only numbers that are clean multiples of 5 or 7.

The Takeaway

In simple terms, this paper is a treasure hunt. The authors took two specific, slightly quirky ways of building number towers (overpartitions and partitions with unique odd parts) and discovered that the total weight of the bricks in these towers follows the same magical, rhythmic patterns Ramanujan found in the standard towers.

They didn't just find the rhythm; they showed why it happens using only the classic, foundational tools of number theory, proving that even in these complex, specialized cases, the universe of numbers still sings in a predictable, beautiful tune.

Technical Summary: Sum of Parts in Overpartitions and Partitions Without Repeated Odd Parts

Problem Statement
This paper investigates Ramanujan-type congruences for specific arithmetic statistics derived from two distinct classes of integer partitions: overpartitions and partitions without repeated odd parts. While Ramanujan's classical congruences for the unrestricted partition function $p(n)$ are well-established, recent work by Andrews and Dastidar extended this inquiry to the sum of odd parts minus the sum of even parts in standard partitions ($SOME(n)$). The authors aim to generalize these findings by studying the sums of parts classified by parity (even vs. odd) and the presence of overlines (non-overlined parts) within overpartitions, as well as the sums of even and odd parts in partitions where odd parts do not repeat.

Methodology
The proofs presented in the paper are strictly elementary, relying on classical theta function identities rather than advanced modular form theory. The core methodology involves the following steps:

Generating Functions: The authors construct generating functions for the specific statistics of interest (e.g., $SNO(n)$, the sum of non-overlined parts in overpartitions). These functions are expressed in terms of infinite products involving $q$ -series, such as $(q; q)_\infty$ and $(-q; q)_\infty$ .
Differentiation and Operators: To extract the sum of parts, the authors utilize differentiation with respect to a parameter $z$ (evaluated at $z=1$ ) or the operator $q \frac{d}{dq}$ . This transforms the generating functions into expressions involving the function $B(q) = \sum \frac{q^n}{(1-q^n)^2}$ .
Dissection Techniques: The central technical tool is the $p$ $p$ -dissection of generating functions for primes $p=5$ $p = 5$ and $p=7$ $p = 7$ . The authors employ:
- Euler's pentagonal number theorem.
- Jacobi's triple product identity.
- Watson's quintuple product identity.
- Weierstrass's identity for products of four theta functions.
Modular Arithmetic: By applying the operator $U_{p,a}$ (which extracts coefficients $f(pn+a)$) and utilizing the dissection formulas modulo $p$ , the authors demonstrate that specific coefficients in the generating functions vanish modulo 5 or 7.

Key Contributions and Results
The paper establishes several new congruences for the defined statistics. The main results are summarized as follows:

Overpartitions:
- Let $SNO(n)$ be the sum of all non-overlined parts in overpartitions of $n$ $n$ . The authors prove:
  - $SNO(5n + 4) \equiv 0 \pmod 5$
  - $SNO(5n + 2) \equiv 0 \pmod 5$
  - $SNO(7n + 3) \equiv 0 \pmod 7$
- Let $SENO(n)$ and $SONO(n)$ denote the sums of even and odd non-overlined parts, respectively. The authors prove:
  - $SENO(7n + 5) \equiv 0 \pmod 7$
  - $SONO(5n + 3) \equiv 0 \pmod 5$
Partitions Without Repeated Odd Parts:
- Let $SEDO(n)$ and $SODO(n)$ denote the sums of even and odd parts in partitions of $n$ $n$ without repeated odd parts. The authors prove:
  - $SEDO(5n) \equiv 0 \pmod 5$ and $SEDO(7n) \equiv 0 \pmod 7$
  - $SODO(5n) \equiv 0 \pmod 5$
  - $SODO(7n) \equiv 0 \pmod 7$ and $SODO(7n + 2) \equiv 0 \pmod 7$
- Furthermore, the difference between the sum of even and odd parts satisfies:
  - $SEDO(7n + 5) - SODO(7n + 5) \equiv 0 \pmod 7$

Significance
The paper claims significance primarily through its methodological approach. By providing elementary proofs for these congruences, the authors demonstrate that deep number-theoretic results regarding partition statistics can be derived using classical theta function identities and dissection techniques, without recourse to more complex machinery. The work extends the recent line of inquiry initiated by Andrews and Dastidar regarding the arithmetic properties of sums of parts, specifically adapting these investigations to the contexts of overpartitions and restricted partitions (those without repeated odd parts). The results contribute to the broader understanding of the distribution of partition statistics modulo small primes.

The Main Discovery

How They Did It (The "Elementary" Magic)

The Takeaway

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