Polynomization of Sun's Conjecture

TL;DR

The paper develops a uniform framework that reinterprets Sun's conjecture on the decreasing sequence $\sqrt[n]{p(n)}$ as a zeros-location problem for generalized D'Arcais polynomials $P_n^g(x)$ via $\Delta_n^g(x)$. By analyzing the largest real zero $x_n^g$ of $\Delta_n^g(x)$ across various $g$, it derives bounds and monotonicity results for partition-related sequences, including $p(n)$, plane partitions, and overpartitions, and relates these to Laguerre and Pochhammer polynomials. It extends the approach to $k$-colored partitions and connects to recent log-concavity results (e.g., BKRT21), yielding new Sun-type inequalities and partial proofs for broader families. The work offers evidence for a broader principle: the sign of $\Delta_n^g(x)$ and the location of $x_n^g$ control monotonicity of $n$th roots across combinatorial sequences, providing a path toward unified proofs and new conjectures in partition theory and related enumerative combinatorics. The significance lies in providing a conceptual, zeros-based, and uniform methodology to derive monotonicity results across diverse partition-like objects, potentially impacting combinatorics, special functions, and statistical mechanics through the NO-polynomials framework.

Abstract

Let $p(n)$ denote the number of partitions of a natural number $n$. As $ n \to \infty$, the $n$th root of $p(n)$ tends to $1$, which is related to the Cauchy--Hadamard test for power series. Andrews also discovered an elementary proof. Sun conjectured that this happens in a certain way for $n\geq 6$: \begin{equation*} \sqrt[n]{p(n)} > \sqrt[n+1]{p(n+1)}. \end{equation*} The conjecture was proved by Wang and Zhu; shortly thereafter, Chen and Zheng independently obtained a second proof. In this paper, we follow an approach by Rota. We consider $p(n)$ as special values of the D'Arcais polynomials, known as the Nekrasov--Okounkov polynomials. This identifies Sun's conjecture as a property of the largest real zero of certain polynomials. This leads to results towards $k$-coloured partitions, overpartitions, and plane partitions. Moreover, we also consider Chebyshev and Laguerre polynomials. The main purpose of this paper is to offer a uniform approach.

Figures (8)

• Figure 1: Largest real zeros of $\Delta_n^{\sigma}(x)$ for $1 \leq n \leq 25$
• Figure 2: Representation of plane partitions
• Figure 3: Largest real zeros of $\left( P_{n}^{\sigma_2}\left( x\right) \right) ^{n+1}-\left( P_{n+1}^{\sigma_2}\left( x\right) \right) ^{n}$ with $n$ vertical
• Figure 4: Largest real zeros of $\Delta_n^{\bar{g}}(x)$ for $1 \leq n \leq 25$
• Figure 5: Largest real zeros of $\Delta_n^{\psi_1}(x)$ for $1 \leq n \leq 25$

Theorems & Definitions (29)

• Theorem 1.1
• Theorem 1.2
• Conjecture 1
• Theorem 1.3
• Theorem 1.4
• Proposition 2.1
• proof
• Lemma 2.2
• proof
• Remark