Large Deviations for the d'Arcais Numbers

Technical Summary: Large Deviations for the d'Arcais Numbers

Problem Statement
This paper investigates the asymptotic behavior of the coefficients of the d'Arcais polynomials, denoted $P_n(z)$. These polynomials are defined via the generating function:
$$\sum_{n=0}^{\infty} P_n(z) q^n = \exp\left(-z \ln((q; q)_\infty)\right)$$
where $(q; q)_\infty = \prod_{k=1}^\infty (1-q^k)$ is the $q$-Pochhammer symbol. The coefficients are given by $P_n(z) = \sum_{k=1}^n A(2, n, k) z^k / n!$. The primary object of study is the normalized coefficient sequence $a(n, k) = k! A(2, n, k) / n!$.

The work is motivated by recent results of Abdesselam regarding the number of commuting $\ell$-tuples of permutations of $[n]$ with a fixed orbit size. While Abdesselam established a Local Central Limit Theorem (LCLT) and log-concavity for the "typicality region" (where the orbit size scales as $x\sqrt{n}$), this paper focuses on the large deviation regime. Specifically, it analyzes the asymptotics of $a(n, k)$ as $n \to \infty$ where the ratio $k_n/n \to \kappa \in [0, 1)$, assuming $k_n \to \infty$.

Methodology
The proof relies on the saddle point method (specifically the Hayman version) applied to Cauchy's integral formula. The authors utilize the generating function structure of the d'Arcais numbers, which are identified as the Bell transform of the abundancy index $\sigma(n)/n$.

Key technical steps include:

1. Generating Function Analysis: The authors define a function $F(y) = -\ln((e^{-y}; e^{-y})_\infty)$, which relates to the generating function of the abundancy index. They analyze the properties of $F(y)$, its derivatives, and its modular symmetry properties derived from Dedekind's eta function.
2. Contour Integration: The coefficient $a(n, k)$ is expressed as a contour integral:
   $$a(n, k) = \frac{1}{2\pi i} \oint z^{-n} \exp\left(k \ln(-\ln((z; z)_\infty))\right) dz$$
   The integration path is a circle of radius $e^{-y}$, where $y$ is chosen to satisfy a saddle point equation.
3. Major and Minor Arcs: Following the Hardy-Ramanujan circle method, the integral is split into a "major arc" (a small neighborhood of $\theta=0$) and a "minor arc."
   • Major Arc: The integrand is expanded using Taylor series. The authors establish uniform bounds for the coefficients and error terms as $y \to 0^+$, utilizing the modular symmetry of $F(y)$ to handle the singularity at $y=0$.
   • Minor Arc: Uniform bounds are derived to show that the contribution from the minor arc is exponentially small compared to the major arc. This involves proving a lower bound on $1 - |F(y-i\theta)|^2/F(y)^2$ using the probability measure $\rho_n(y)$ associated with the abundancy index.
4. Legendre-Fenchel Transform: The large deviation rate function is identified as the Legendre-Fenchel transform of a specific function $g$, derived from the inverse of $f(y) = \ln(F(y))$.

Key Contributions and Results

1. Bahadur-Rao Type Large Deviation Formula:
   The main result (Theorem 3.1) provides an asymptotic formula for $a(n, k_n)$ as $n \to \infty$ with $k_n/n \to \kappa \in [0, 1)$ and $k_n \to \infty$:
   $$\frac{k_n! A(2, n, k_n)}{n!} \sim \frac{\exp(k_n \ln(F(y_n)) + n y_n)}{\sqrt{2\pi k_n \left( \frac{F''(y_n)}{F(y_n)} - \left(\frac{F'(y_n)}{F(y_n)}\right)^2 \right)}}$$
   Here, $y_n$ is the unique solution to $-F'(y_n)/F(y_n) = 1/\kappa_n$ (where $\kappa_n = k_n/n$).

2. Log-Concavity in the Large Deviation Regime:
   Corollary 3.2 establishes that the sequence $a(n, k)$ is log-concave for sufficiently large $n$ in this regime. Specifically, the ratio $\frac{a(n, k)^2}{a(n, k-1)a(n, k+1)}$ is shown to be positive and asymptotically equivalent to a specific function of $y_n$ and $n$. This extends the log-concavity results previously known only for the typicality region ($k \sim \sqrt{n}$) to the large deviation region.

3. Asymmetry of the Rate Function:
   The paper proves that the large deviation rate function is not symmetric. This is demonstrated by showing that the function $F(y)F(y + \ln(F(y)))$ is not identically equal to 1, a condition that would be necessary for symmetry. The asymmetry is linked to the specific modular properties of the Dedekind eta function.

4. Connection to Abundancy Index:
   The authors explicitly relate the d'Arcais numbers to the Bell transform of the abundancy index $\sigma(n)/n$. The asymptotic analysis relies heavily on the analytic properties of the generating function for the abundancy index, which is $F(y)$.

Significance and Scope
The paper claims to provide an alternative derivation of log-concavity for the case $\ell=2$ (pairs of commuting permutations) in a regime distinct from Abdesselam's LCLT results. The authors note that for $\ell=2$, the relevant generating function possesses modular symmetry (via Dedekind's eta function), which facilitates the derivation of these asymptotics.

The authors are modest regarding the scope of their results:

• They focus strictly on $\ell=2$. They do not extend the results to $\ell > 2$, noting that while Abdesselam has results for the typicality region for higher $\ell$, the large deviation regime for $\ell > 2$ remains an open problem requiring further understanding of potential quasimodularity properties.
• They acknowledge that their results cover the region where $k/n \to \kappa < 1$, leaving the boundary case where $n-k$ is order 1 (covered by Tripathi) and the region where $k/n \to 1$ as outside the current scope.
• The work serves as a self-contained application of the circle method without Farey fractions, contrasting with the more complex apparatus required for the abundancy index directly, and serves as a warm-up for the Hardy-Ramanujan partition formula in the appendix.

In summary, the paper establishes precise large deviation asymptotics for the d'Arcais numbers, proves log-concavity in this regime, and highlights the role of modular symmetry in these combinatorial problems, while deferring the generalization to higher-order commuting tuples to future work.