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Asymptotic Formula for Multipartitions

This paper derives an asymptotic formula for the number of $t$ -multipartitions of a positive integer $N$ in the regime where the number of parts $t$ is significantly smaller than $N^{1-\epsilon}$ for any $\epsilon > 0$ .

Original authors: Jayanta Barman, Kamalakshya Mahatab

Published 2026-07-13

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Jayanta Barman, Kamalakshya Mahatab

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 bag of identical Lego bricks. Your job is to build towers using exactly $N$ bricks. In the world of math, this is called a "partition." You can stack them in a single tower, or you can split them into two separate towers, or three, or even more.

Now, imagine you have a special set of instructions called $t$ -multipartitions. This is like saying, "You must build exactly $t$ different towers, and the total number of bricks across all of them must equal $N$ ." If $t=2$ , you're building two towers. If $t=5$ , you're building five. The question mathematicians have been asking for a long time is: How many different ways can you arrange these towers for a huge number of bricks?

For a very long time, we only knew the answer when the number of towers ( $t$ ) was a small, fixed number, like 1, 2, or 10. It was like having a recipe that worked perfectly for a small cake but failed when you tried to bake a skyscraper-sized one.

The Big Discovery
In this paper, Jayanta Barman and Kamalakshya Mahatab have baked a new, much bigger recipe. They found a way to calculate the number of ways to build these $t$ towers, even when the number of towers ( $t$ ) is huge—specifically, when $t$ is smaller than $N$ raised to a power slightly less than 1 (written as $t \ll N^{1-\epsilon}$ ).

Think of it this way: If you have a million bricks ( $N$ ), previous recipes could only tell you how to count the arrangements if you were building a handful of towers. These authors figured out how to count the arrangements even if you were building thousands of towers, as long as you aren't trying to build a tower for every single brick.

How They Did It: The Saddle Point Trick
To solve this, the authors didn't just count one by one (which would take forever). Instead, they used a mathematical tool called the Saddle Point Method.

Imagine the number of ways to build your towers as a giant, rolling mountain range. Most of the time, the terrain is flat and boring, but there is one specific spot—the "saddle point"—where the mountain dips just right. The authors realized that almost all the possible ways to arrange your towers are concentrated right around this one special spot.

They used a clever balancing act to find exactly where this saddle point is. They set up an equation where two forces balance each other out: one force related to the number of bricks ( $N$ ) and another related to the number of towers ( $t$ ). By finding the exact spot where these forces cancel out, they could zoom in on that tiny area and count the arrangements with incredible precision.

What They Found (and What They Didn't)
The authors proved that for any tiny margin of error you want (let's call it $\epsilon$ ), their formula works perfectly as long as the number of towers isn't too close to the number of bricks.

Their final formula looks a bit like a complex magic spell, but it tells you exactly how the number of arrangements grows. It involves:

A base number related to the towers: $\left(\frac{t}{24}\right)^{\frac{t+1}{4}}$
An exponential explosion of possibilities: $\exp\left(2\pi\sqrt{6}\sqrt{t}\sqrt{N - \frac{t}{24}}\right)$
A correction factor to make it precise: $\frac{1}{\sqrt{2}\left(N - \frac{t}{24}\right)^{\frac{t+3}{4}}}$

They also showed that if you set $t=1$ (just one tower), their magic spell turns into the famous formula discovered by Hardy and Ramanujan over a century ago. If you keep $t$ fixed at any small number, their formula matches a result by Murty from 2015. This proves their new, giant recipe is consistent with all the old, trusted ones.

How Sure Are They?
The authors didn't just guess or simulate this on a computer; they proved it mathematically. They used rigorous steps to show that their formula is an "asymptotic" truth. This means that as the number of bricks ( $N$ ) gets larger and larger, their formula gets closer and closer to the real answer, with a tiny, predictable error that shrinks as the numbers grow.

They explicitly ruled out the idea that this only works for fixed, small numbers of towers. They showed it works for a dynamic range where the number of towers can grow along with the number of bricks, as long as the towers don't outnumber the bricks too closely.

So, if you ever find yourself with a mountain of Lego bricks and a demand to build thousands of towers, you now know there is a precise mathematical map to count every single possibility, thanks to this new saddle-point journey.

Technical Summary: Asymptotic Formula for Multipartitions

Problem Statement
The paper addresses the asymptotic behavior of $p_t(N)$ , the number of $t$ -multipartitions of a positive integer $N$ . A $t$ -multipartition is defined as a sequence of $t$ integer partitions whose parts sum to $N$ . Equivalently, this counts the number of $t$ -colored partitions of $N$ . While the asymptotic theory for the ordinary partition function $p(N)$ (where $t=1$ ) is well-established via the work of Hardy, Ramanujan, and Rademacher, and Murty [15] previously derived an asymptotic formula for $p_t(N)$ for a fixed $t$ , a gap remained in the analysis when $t$ varies with $N$ . Specifically, the authors aim to derive an asymptotic formula for $p_t(N)$ in the range where $t \ll N^{1-\epsilon}$ for any $\epsilon > 0$ , rather than restricting $t$ to be a constant.

Methodology
The authors employ the saddle point method applied to the generating function of $p_t(N)$ . The generating function is given by:
$G(q) = \prod_{N=1}^{\infty} (1-q^N)^{-t} = \frac{q^{t/24}}{\eta(z)^t}$
where $q = e^{2\pi i z}$ and $\eta(z)$ is the Dedekind eta function. Using the Cauchy integral formula, $p_t(N)$ is expressed as a contour integral. The authors define a saddle point $y > 0$ (where $z = iy$) determined by the equation:
$y^2 M + t\mu_1(iy) = 0$
where $M = N - t/24$ and $\mu_m(z)$ are auxiliary functions related to the derivatives of $\log \eta(z)$ .

The core of the analysis involves decomposing the integral into two regions based on the real part of $z$ (denoted $x$ ):

The Central Region ( $|x| < y/3$ ): Here, the integrand is expanded using a Taylor series of $\log \eta(z)$ around the imaginary axis. The authors utilize explicit formulas for $\mu_m(z)$ for small imaginary parts (derived from the functional equation of the eta function) to approximate the integral.
The Tail Region ( $y/3 \le |x| \le 1/2$ ): In this region, the Taylor expansion is not valid. The authors establish an upper bound for the ratio $|\eta(iy)/\eta(z)|^t$ using modular transformation properties and the functional equation of the eta function, showing that this region contributes only to the error term.

Key technical tools include:

Lemma 3.1: An approximation for $\eta(iy)$ for small $y$ .
Lemma 3.2: An exponential decay bound for the integrand in the tail region.
Lemma 3.3: An explicit integral evaluation formula involving Gaussian-like terms.
Lemma 3.6: Bounds on the ratios of the auxiliary functions $\mu_3$ and $\mu_4$ relative to $\mu_2$ .

Key Results
The paper establishes two primary theorems:

Theorem 1.1: It proves the existence of a unique solution $y > 0$ to the saddle point equation. It provides an asymptotic approximation for $p_t(N)$ in terms of this $y$ :
$p_t(N) = \frac{y^{3/2} \exp(2\pi M y) a_t(iy)}{\sqrt{t \mu_2(iy)}} \left(1 + O\left(\frac{y}{t}\right)\right)$
where $a_t(z) = \eta(z)^{-t}$ .
Theorem 1.2: By explicitly solving for $y$ in terms of $N$ and $t$ under the condition $t \ll N^{1-\epsilon}$ , the authors derive a concrete asymptotic formula:
$p_t(N) = \left(\frac{t}{24}\right)^{\frac{t+1}{4}} \frac{\exp\left(2\pi\sqrt{6}\sqrt{t}\sqrt{N - \frac{t}{24}}\right)}{\sqrt{2}\left(N - \frac{t}{24}\right)^{\frac{t+3}{4}}} \exp\left(O\left(\frac{t^{3/2}}{\sqrt{N}}\right)\right) \left(1 + O\left(\frac{1}{\sqrt{Nt}}\right)\right)$

Corollaries and Consistency
The authors verify that their results align with known literature:

Setting $t=1$ in Theorem 1.2 recovers Rademacher's asymptotic formula for the ordinary partition function $p(N)$ .
Setting $t$ to be a fixed positive integer recovers Murty's result [15], including the specific error term structure.
The result is compared with Meinardus's formula [14], demonstrating consistency with the general theory of partition functions with constraints.

Significance
The paper claims to extend the asymptotic analysis of multipartitions from the case of fixed $t$ to the more general regime where $t$ grows with $N$ (specifically $t \ll N^{1-\epsilon}$ ). By utilizing the saddle point method with careful error analysis in both the central and tail regions of the contour integral, the authors provide a rigorous asymptotic formula that includes explicit error terms. This generalization is significant for understanding the enumerative properties of multipartitions in a broader arithmetic context, bridging the gap between fixed-parameter results and variable-parameter regimes in partition theory.

Technical Summary: Asymptotic Formula for Multipartitions

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