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Recurrence and congruences for the smallest parts function

This paper establishes generalized Euler-like recursive formulas for the smallest parts function $\spt(n)$ using Hecke traces of twisted quadratic Dirichlet series, derives a closed-form expression for its power series modulo $\ell$ in terms of cusp forms, and proves a new incongruence result.

Original authors: Wei Wang

Published 2026-04-16

📖 5 min read🧠 Deep dive

CC0 1.0

Original authors: Wei Wang

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ 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. You want to know how many different ways you can build a tower using exactly $n$ bricks. In the world of mathematics, this is called the partition function, denoted as $p(n)$ . It's a classic puzzle that has fascinated mathematicians for centuries.

But this paper isn't just about counting towers; it's about a specific, slightly more complicated version of the puzzle called the Smallest Parts Function, or spt(n).

The Core Concept: The "Smallest Brick" Rule

Let's say you build a tower of 5 bricks. You could stack them all in a single column (5), or make a base of 4 with one on top (4+1), or a base of 3 with two on top (3+2), and so on.

In every single one of these towers, there is a smallest brick (or a group of identical smallest bricks).

In the tower "3+1+1", the smallest brick is the "1". There are two of them.
In the tower "2+2+1", the smallest is "1". There is one of them.

The spt(n) function simply counts the total number of these "smallest bricks" across all possible towers you can build with $n$ bricks.

The Problem: A Messy, Non-Stop Equation

For a long time, mathematicians knew how to calculate $p(n)$ (the total towers) using a clever, repeating recipe called a "recurrence relation." It's like a recipe that says, "To make the next batch of cookies, take the previous batch, subtract a few, add a few, and you're done."

However, the recipe for spt(n) was much messier. It didn't follow the same clean rules. The generating function (the mathematical machine that spits out these numbers) was "broken" or "non-holomorphic." Imagine trying to bake a cake where the batter keeps changing its texture halfway through the mixing process. It's hard to predict the final result.

The Solution: The "Shadow" and the "Projection"

The author, Wei Wang, uses a sophisticated mathematical technique called Holomorphic Projection.

Think of the messy spt(n) function as a shadow cast by a 3D object onto a 2D wall. The shadow is distorted and hard to measure directly. But if you know exactly where the light source is and the shape of the object, you can mathematically "project" the shadow back onto a clean, flat surface where it makes perfect sense.

In this paper:

The Object: A "Harmonic Maass Form." This is a complex, multi-dimensional mathematical shape that contains the spt(n) data but is too wobbly to work with directly.
The Projection: Wang uses a special mathematical "lens" (holomorphic projection) to flatten this wobbly shape into a clean, solid Modular Form. This is like turning that distorted shadow into a crisp, clear blueprint.

The New Recipe: Hecke Traces

Once the blueprint is clean, Wang discovers a new, powerful recipe (a recurrence relation) for spt(n).

Old Recipe: Just subtract and add numbers based on "pentagonal numbers" (a specific pattern of shapes).
New Recipe: To find the next number, you don't just look at the past numbers. You have to look at a "Hecke Trace."

What is a Hecke Trace?
Imagine a choir of singers (these are the "Hecke eigenforms"). Each singer has a unique voice (a number sequence). A "Hecke Trace" is like taking the average volume of all the singers at a specific moment in the song. Wang's new recipe says: "To find the next spt number, look at the average volume of this specific choir of mathematical singers."

This connects the humble counting of LEGO bricks to deep, high-level symphonies of number theory.

The Magic of Patterns (Congruences)

The paper also proves some "magic tricks" about these numbers.

Ramanujan, a legendary mathematician, discovered that if you look at the total number of towers ( $p(n)$ ) for numbers like 5, 10, 15... (specifically $5n+4$ ), the answer is always divisible by 5. It's a hidden pattern.

Wang proves that spt(n) has similar magic tricks.

If you look at $spt(5n+4)$, the answer is always divisible by 5.
If you look at $spt(7n+5)$, the answer is always divisible by 7.

But here is the twist: Wang also proves that these patterns break for certain other numbers. He shows that for many primes (like 13, 17, etc.), the "magic trick" fails. The numbers don't line up perfectly. This is like finding a rule that works for every Tuesday but fails on every Thursday.

Why Does This Matter?

It Unifies Ideas: It connects a simple counting problem (LEGO towers) with the most advanced tools in modern mathematics (Modular Forms and Maass Forms).
It Solves a Mystery: It gives us a precise, step-by-step way to calculate these numbers without having to list every single tower.
It Reveals Limits: By proving where the patterns don't work, it helps mathematicians understand the boundaries of these mathematical structures.

In a Nutshell

Wei Wang took a messy, hard-to-calculate number sequence (spt), used a mathematical "lens" to clean it up, and discovered a new, elegant way to predict its future values by listening to the "song" of other deep mathematical objects. He also mapped out exactly where the hidden patterns of these numbers hold true and where they break, adding a new chapter to the story of how numbers behave.

1. Problem Statement

The paper addresses the arithmetic properties of the smallest parts function, denoted as $spt(n)$. Introduced by Andrews, $spt(n)$ counts the total number of smallest parts in all partitions of an integer $n$ .

Context: While the partition function $p(n)$ has well-known recurrence relations (Euler's recurrence) and Ramanujan-type congruences, the $spt(n)$ function is more complex. Its generating function is not a weakly holomorphic modular form but a harmonic Maass form.
Goal: The author aims to:
1. Generalize Euler-like recurrence formulas for $spt(n)$ using the theory of modular forms and Hecke traces, analogous to recent generalizations for $p(n)$ .
2. Establish closed-form expressions for power series of $spt(n)$ modulo primes $\ell$ .
3. Prove incongruence results (showing when $spt(n)$ does not vanish modulo $\ell$ ).

2. Methodology

The paper employs advanced techniques from the theory of modular forms, specifically focusing on harmonic Maass forms and holomorphic projection.

Harmonic Maass Forms: The generating function for $spt(n)$, denoted $\alpha(z)$ , is part of a harmonic Maass form $F(z)$ of weight $3/2$ . Unlike standard modular forms, $F(z)$ has a non-holomorphic part involving incomplete gamma functions.
Holomorphic Projection: To extract arithmetic information, the author applies the holomorphic projection operator ( $\pi_{hol}$ ) to the product of the harmonic Maass form and the Dedekind eta function ( $\eta$ ). This maps the non-holomorphic object into the space of holomorphic modular forms.
Rankin-Cohen Brackets: The author utilizes Rankin-Cohen brackets $[f, g]_v$ to construct modular forms of higher weights ( $2v+2$ ) from the weight $3/2$ form and the weight $1/2$ eta function.
Maass-Poincaré Series: A significant technical hurdle is that the standard Maass-Poincaré series used for $p(n)$ fails to converge for the $spt$ context. The author constructs Maass-Poincaré series with the eta multiplier, proves their analytic continuation to the critical point $s=3/4$ , and identifies them with the generating function of $spt(n)$.
Hecke Traces: The core of the recurrence relations involves Hecke traces of special twisted quadratic Dirichlet series associated with cusp forms.

3. Key Contributions and Results

A. Generalized Euler-like Recurrence (Theorem 1 & Corollary 2)

The paper derives a generalized recurrence relation for $spt(n)$ that mirrors the structure of the partition function recurrence but includes terms related to Hecke traces.

Main Identity: For every $v \geq 1$ , the author defines a polynomial $q_v(n, k)$ (related to Chebyshev polynomials) and a series $S_v(z)$ . The main theorem states:
$S_v(z) = E_{2v+2}(z) - 12F_{2v+2}(z) + \text{Hecke Trace Term}$
where $E_{2v+2}$ is the Eisenstein series, $F_{2v+2}$ is a mock Eisenstein series, and the last term involves the sum over normalized Hecke eigenforms $f$ of weight $2v+2$ .
Explicit Recurrence: By examining coefficients, the author provides an explicit recurrence for $s(n) = 12spt(n) + (24n-1)p(n)$:
$q_v(n, 0)s(n) = \sum_{k \neq 0} (-1)^{k+1} q_v(n, k)s(n-\omega(k)) + \text{Divisor Terms} - \text{Hecke Trace Term}$
This allows for the computation of $spt(n)$ using divisor functions and traces of cusp forms.

B. Closed Forms Modulo $\ell$ (Corollary 3)

The paper establishes a closed-form expression for the generating function of $spt(\ell n - \delta_\ell)$ modulo a prime $\ell \geq 5$ , where $\delta_\ell = (\ell^2-1)/24$ .

Result:
$\sum_{n \geq 1} spt(\ell n - \delta_\ell) q^n \equiv 12^{-1} \left(\frac{3}{\ell}\right) \frac{\sum_{n \geq 1} \text{Tr}_{\ell+1}(\ell n) q^n}{(q^\ell; q^\ell)_\infty} \pmod \ell$
Significance: This provides a new proof of Andrews' congruences for $spt(5n+4)$, $spt(7n+5)$, and $spt(13n+6)$ by noting the non-existence of non-trivial cusp forms for weights 6, 8, and 14. It also yields new congruences for other primes (e.g., $\ell=11$ ).

C. Incongruence Results (Corollary 4)

The paper addresses the conjecture that simple congruences $spt(\ell n - \delta_\ell) \equiv 0 \pmod \ell$ fail for most primes.

Result: Using a theorem by Garthwaite and Jameson, the author proves that if a specific Legendre symbol condition holds and $spt(t_0) \not\equiv 0$ , then the series $\sum spt(\ell n + t)q^n$ is not identically zero modulo $\ell$ for certain $t$ . This confirms that the "Ramanujan-like" congruences are rare and do not hold universally for $spt(n)$.

4. Technical Significance

Overcoming Convergence Issues: The paper successfully adapts the Maass-Poincaré series method to the $spt$ function, a case where standard convergence fails, by utilizing analytic continuation. This is a novel application of the technique.
Connection to Mock Modular Forms: It rigorously connects the combinatorial $spt$ function to the theory of mock modular forms and harmonic Maass forms, specifically through the holomorphic projection of the product $F \cdot \eta$ .
Arithmetic Depth: The results deepen the understanding of the arithmetic nature of $spt(n)$, linking it directly to the Hecke traces of cusp forms, similar to how $p(n)$ is linked to traces in the work of Gomez, Ono, et al.
New Proofs: The paper provides alternative, modular-form-based proofs for known congruences (Andrews' congruences) and extends them to new cases (e.g., $\ell=11$ ).

5. Conclusion

Wei Wang's paper successfully generalizes Euler's recurrence for the partition function to the smallest parts function $spt(n)$. By navigating the complexities of harmonic Maass forms and establishing a rigorous link between $spt(n)$ and Hecke traces of cusp forms, the author derives powerful recurrence relations and congruence properties. These results not only unify the theory of partitions and smallest parts but also provide new tools for investigating the arithmetic distribution of $spt(n)$ modulo primes.