Inequalities for the number of $t$-hooks in two partition classes arising from sum-product identities

1. Problem Statement and Motivation

The paper investigates the distribution of $t$-hooks in integer partitions, specifically comparing two distinct classes of partitions defined by sum-product identities.

• Background: The study is motivated by recent work on Euler's partition identity (distinct parts vs. odd parts). Andrews proved that for $n \ge 1$, the total number of parts in partitions of $n$ into distinct parts ($b_1(n)$) is greater than or equal to the total number of different parts in partitions of $n$ into odd parts ($a_1(n)$). However, for $t > 1$, the inequality reverses for large $n$ (i.e., $a_t(n) > b_t(n)$), a conjecture recently proven by Craig, Dawsey, and Han.
• The Core Question: The authors extend this "hook bias" phenomenon to other partition identities, specifically those involving gap conditions (parts differing by at least $k$) versus congruence conditions (parts satisfying specific modular arithmetic).
• Specific Targets: The paper focuses on two major identities:
  1. The First Rogers–Ramanujan Identity: Comparing partitions with parts differing by at least 2 ($R_1$) against partitions with parts congruent to $1, 4 \pmod 5$ ($R_2$).
  2. The First Little Göllnitz Identity: Comparing partitions with parts differing by at least 2 (with no odd parts differing by exactly 2) ($G_1$) against partitions with parts congruent to $1, 5, 6 \pmod 8$ ($G_2$).

The goal is to determine the asymptotic behavior of $r_{j,t}(n)$ and $g_{j,t}(n)$ (the total number of $t$-hooks in the respective sets) and establish inequalities for $t=1$ and $t=2$.

2. Methodology

The authors employ a sophisticated analytic number theory approach combining generating functions, asymptotic analysis, and Tauberian theorems.

A. Generating Functions

The first step involves deriving bivariate generating functions $R_{j,t}(x, q)$ and $G_{j,t}(x, q)$, where the variable $x$ marks the number of $t$-hooks.

• The generating function for the number of $t$-hooks, $S_{j,t}(q)$, is obtained by differentiating with respect to $x$ at $x=1$:
  $$S_{j,t}(q) = \frac{\partial}{\partial x} R_{j,t}(x, q) \bigg|_{x=1}$$
• For $t=1$, the number of hooks corresponds to the number of parts (or distinct parts), leading to relatively simple derivatives.
• For $t=2$, the analysis requires decomposing partitions based on specific part sizes and multiplicities, resulting in more complex expressions involving infinite products and rational functions.
• The resulting generating functions are identified as Nahm sums (q-series where the ratio of consecutive coefficients is rational) or products of infinite products and rational functions.

B. Asymptotic Analysis (Saddle Point Method)

To find the asymptotic behavior of the coefficients $r_{j,t}(n)$ and $g_{j,t}(n)$, the authors analyze the generating functions as $q \to 1$ (specifically $q = e^{-z}$ where $z = \epsilon(1+iy)$ and $\epsilon \to 0^+$).

1. Peak Identification: They identify the "peak" of the summand in the generating function series. For Rogers–Ramanujan, the peak occurs near $n \approx \frac{\log \phi}{\epsilon}$ (where $\phi$ is the golden ratio). For Little Göllnitz, it occurs near $n \approx \frac{\log(\sqrt{2}+1)}{2\epsilon}$.
2. Decomposition: The sum is split into a "main" part (near the peak, $N_\epsilon$) and an "error" part (far from the peak).
3. Local Expansion: Using Lemma 2.1 (an extension of Zagier's result on the asymptotic expansion of the q-Pochhammer symbol) and the Euler-Maclaurin summation formula, they derive precise asymptotic expansions for the summands near the peak.
4. Saddle Point Integration: The sum near the peak is approximated by a Gaussian integral, yielding the leading asymptotic term.
5. Error Control: They rigorously prove that the contribution from the "tail" (far from the peak) is negligible compared to the main term, even when $y$ (the imaginary part of $z$) varies within a bounded cone. This involves analyzing the real part of the exponent $\Lambda(y)$ to ensure it is maximized at $y=0$.

C. Tauberian Theorem

Once the asymptotic behavior of the generating function $S(q) \sim \alpha z^{-\beta} e^{\gamma/z}$ is established as $z \to 0$, the authors apply Ingham's Tauberian Theorem (Theorem 2.4).

• Since the coefficients (hook counts) are weakly increasing, the theorem allows them to translate the asymptotic behavior of the generating function directly into the asymptotic behavior of the coefficients $a_n$.
• The formula used is:
  $$a_n \sim \frac{\alpha \gamma^{\frac{\beta}{2} + \frac{1}{4}}}{2\sqrt{\pi} n^{\frac{\beta}{2} + \frac{3}{4}}} e^{2\sqrt{\gamma n}}$$

3. Key Contributions and Results

A. Rogers–Ramanujan Identities

The authors derive the following asymptotic formulas as $n \to \infty$:

• For $t=1$:
  $$r_{1,1}(n) \sim r_{2,1}(n) \sim \frac{3^{1/4} \phi^{1/2} \log(\phi)}{2\pi} n^{-1/4} e^{2\pi \sqrt{n/15}}$$
  Result: $r_{1,1}(n) > r_{2,1}(n)$ for sufficiently large $n$. (The gap condition set has more 1-hooks).
• For $t=2$:
  $$r_{1,2}(n) \sim \frac{3^{1/4} \phi^{1/2}}{5\pi} n^{-1/4} e^{2\pi \sqrt{n/15}}$$
  $$r_{2,2}(n) \sim \frac{3^{5/4} \phi^{1/2}}{10\pi} n^{-1/4} e^{2\pi \sqrt{n/15}}$$
  Result: $r_{1,2}(n) < r_{2,2}(n)$ for sufficiently large $n$. (The congruence condition set has more 2-hooks).

B. Little Göllnitz Identities

Similarly, for the Little Göllnitz sets:

• For $t=1$:
  $$g_{1,1}(n) \sim g_{2,1}(n) \sim \frac{\log(\sqrt{2}+1)}{2^{5/4}\pi} n^{-1/4} e^{\pi \sqrt{n/2}}$$
  Result: $g_{1,1}(n) > g_{2,1}(n)$ for sufficiently large $n$.
• For $t=2$:
  $$g_{1,2}(n) \sim \frac{3 \cdot 2^{1/4}}{13\pi} n^{-1/4} e^{\pi \sqrt{n/2}}$$
  $$g_{2,2}(n) \sim \frac{2^{1/4}}{5\pi} n^{-1/4} e^{\pi \sqrt{n/2}}$$
  Result: $g_{1,2}(n) < g_{2,2}(n)$ for sufficiently large $n$.

4. Significance and Implications

1. Generalization of Hook Bias: The paper confirms that the "hook bias" phenomenon observed in Euler's identity (where distinct parts have more 1-hooks but fewer $t$-hooks for $t>1$ compared to odd parts) is a robust feature that extends to the Rogers–Ramanujan and Little Göllnitz identities.
2. Analytic Technique Refinement: The authors provide a streamlined and detailed proof technique for analyzing Nahm sums and q-series related to partition identities. They simplify previous methods (e.g., those by Bringmann et al.) while maintaining rigorous error bounds, particularly in handling the complex variable $z$ near the unit circle.
3. Conjecture for Higher $t$: Based on the results for $t=1, 2$ and numerical experiments, the authors conjecture that for all $t > 2$, the inequality reverses again, such that the congruence condition partitions have more $t$-hooks than the gap condition partitions ($r_{1,t}(n) < r_{2,t}(n)$ and $g_{1,t}(n) < g_{2,t}(n)$).
4. Future Directions: The paper suggests that these asymptotic methods can be applied to other identities with gap conditions, such as the second Rogers–Ramanujan identity and the Göllnitz–Gordon identities, opening the door for further research into the statistical properties of partition hooks.

In summary, this work bridges combinatorial partition theory with analytic number theory, providing precise asymptotic formulas that reveal a consistent structural bias in the distribution of hooks across different classes of partitions defined by classical identities.