Asymptotic Formula for $(t+1)$-Regular Partitions

1. Problem Statement

The paper addresses the asymptotic behavior of $(t+1)$-regular partitions, denoted as $p(N, t)$.

• Definition: A partition of an integer $N$ is called $t$-regular if none of its parts is divisible by $t$. Consequently, $p(N, t)$ counts the number of partitions of $N$ where no part is divisible by $t+1$.
• Context: While the asymptotic formula for unrestricted partitions $p(N)$ was established by Hardy and Ramanujan (1918) and refined by Rademacher (1938), and Hagis (1971) derived an asymptotic formula for $p(N, t)$ using the circle method for fixed $t$, there was a lack of explicit, rigorous bounds for $p(N, t)$ when $t$ varies relative to $N$ (specifically when $t$ grows with $N$).
• Goal: The authors aim to derive explicit asymptotic formulas for $p(N, t)$ across different ranges of $t$ (relative to $N$) using the saddle point method, providing explicit error bounds and extending Hagis's results.

2. Methodology

The authors employ the saddle point method, a powerful technique in analytic combinatorics for estimating coefficients of generating functions.

• Generating Function: They utilize the generating function for $p(N, t)$:
  $$G(q, t) = \sum_{N=0}^\infty p(N, t)q^N = \frac{F(q)}{F(q^{t+1})} = \frac{\eta((t+1)z)}{\eta(z)}$$
  where $q = e^{2\pi i z}$ and $\eta(z)$ is the Dedekind eta function.
• Integral Representation: Using Cauchy's integral formula, $p(N, t)$ is expressed as an integral over a contour in the complex plane:
  $$p(N, t) = \int_{-1/2}^{1/2} \exp(-2\pi i z L) g_t(z) \, dx$$
  where $L = N + \frac{t}{24}$ and $g_t(z) = \frac{\eta((t+1)z)}{\eta(z)}$.
• Saddle Point Analysis:
  1. Saddle Point Equation: They define a function $a(z) = -2\pi i N z + \log g_t(z)$ and solve $a'(iy) = 0$ to find the optimal saddle point $y > 0$. This leads to the equation involving the functions $\mu_k(z)$ (derivatives of $\log \eta(z)$):
     $$\frac{\mu_1((t+1)iy) - (t+1)\mu_1(iy)}{(t+1)y^2} = N + \frac{t}{24}$$
  2. Region Partitioning: The integration range is split into a "central" region ($|x| < y/3$) where Taylor expansion is valid, and a "tail" region ($y/3 \le |x| \le 1/2$).
  3. Error Control (Tail Region): A critical technical contribution is Proposition 3.1, which bounds the integral in the tail region. Since Taylor expansion fails here, the authors use the modular transformation properties of the Dedekind eta function under the action of $SL_2(\mathbb{Z})$ to derive exponential decay bounds.
  4. Gaussian Approximation: In the central region, the integral is approximated by a Gaussian integral, yielding the main term, while higher-order terms are absorbed into the error bound.

3. Key Contributions and Results

The paper provides explicit asymptotic formulas for $p(N, t)$ in three distinct regimes of $t$:

Regime 1: Small $t$ (Fixed or slowly growing)

• Condition: $(t+1) \le \frac{2\pi}{(\frac{1}{2}+\epsilon)\log(N + t/24)} \sqrt{\frac{24(t+1)(N+t/24)}{t}}$.
• Result: The authors derive a precise asymptotic formula:
  $$p(N, t) \sim \sqrt{12} A_t(N) \exp\left( 2\pi\sqrt{6} \sqrt{\frac{t}{t+1}\left(N+\frac{t}{24}\right)} \right) \times \text{correction terms}$$
  where $A_t(N)$ is an explicit algebraic factor. This refines Hagis's result by providing explicit constants and error terms ($O(N^{-1/2})$).

Regime 2: Intermediate $t$

• Condition: The range between Regime 1 and $t \approx \sqrt{24N}$.
• Result: An upper bound is established for $p(N, t)$, acknowledging the complexity of the modular properties in this transition zone.

Regime 3: Large $t$ ($t > \sqrt{24N}$)

• Condition: $(t+1) > \sqrt{24N}$.
• Result: The behavior of $p(N, t)$ closely approximates the unrestricted partition function $p(N)$. The formula is:
  $$p(N, t) = p(N) \exp\left( -\nu_1 \exp\left( -\frac{\pi(t+1)}{\sqrt{6N}} \right) \right) (1 + O(N^{-1/2}))$$
  This shows that for very large $t$, the restriction of regularity has a negligible effect on the total count.

Special Case: Fixed $t$

• The paper recovers Theorem 1.3, which coincides with Hagis's 1971 formula for fixed $t$, validating the new method.

4. Application: Zeros in Character Tables

A significant application of these results is in the representation theory of the symmetric group $S_N$.

• Problem: Estimating $Z_t(N)$, the number of pairs of partitions $(\lambda, \mu)$ such that the irreducible character $\chi_\lambda(\mu) = 0$, where $\lambda$ is a $t$-core.
• Connection: The number of such zeros is related to the difference between the total number of partitions $p(N)$ and the number of $(t+1)$-regular partitions $p(N, t-1)$.
• Improvement: Using their new asymptotic bounds for $p(N, t)$, the authors generalize and improve upon previous results by McSpirit and Ono. They provide bounds for $Z_t(N)$ that hold for a much wider range of $t$ (up to $t \ll N^{1/2-\delta}$), rather than just fixed $t$.

5. Significance

• Methodological Advancement: The paper successfully transitions the analysis of regular partitions from the circle method (Hagis) to the saddle point method, allowing for more flexible handling of the parameter $t$ as it varies with $N$.
• Explicit Bounds: Unlike many asymptotic results that provide only the leading term, this work provides explicit constants and rigorous error bounds ($O(N^{-1/2})$), making the formulas usable for numerical estimation and further theoretical applications.
• Bridging Fields: It connects analytic number theory (partition asymptotics) with algebraic combinatorics (character tables of symmetric groups), offering new insights into the distribution of zeros in character tables.
• Completeness: By covering the entire spectrum of $t$ (from small to large relative to $N$), the paper provides a comprehensive picture of how the regularity constraint affects partition growth.