Related papers: Polynomization of Sun's Conjecture

Nicolas and DeSalvo and Pak proved that the partition function $p(n)$ is log concave for $n \geq 25$. Chen, Jia and Wang proved that $p(n)$ satisfies the third order Tur\'{a}n inequality, and that the associated degree 3 Jensen polynomials…

Let $n>1$ be an odd integer. For any primitive $n$-th root $\zeta$ of unity in the complex field. Via the Engenvector-eigenvalue Identity, we show that $$\sum_{\tau\in…

The 2-color partitions may be considered as an extension of regular partitions of a natural number $n$, with $p_{k}(n)$ defined as the number of 2-colored partitions of $n$ where one of the 2 colors appears only in parts that are multiples…

In the $1970$s, Nicolas proved that the partition function $p(n)$ is log-concave for $ n > 25$. In \cite{HNT21}, a precise conjecture on the log-concavity for the plane partition function $\func{pp}(n)$ for $n >11$ was stated. This was…

Let $p\leq 23$ be a prime and $a_p(n)$ counts the number of partitions of $n$ where parts that are multiple of $p$ come up with $2$ colors. Using a result of Sussman, we derive the exact formula for $a_p(n)$ and obtain an asymptotic formula…

Let $p_{\{3, 3\}}(n)$ denote the number of $3$-regular partitions in three colours. In a very recent paper, da Silva and Sellers studied certain arithmetic properties of $p_{\{3, 3\}}(n)$. They further conjectured four Ramanujan-like…

In this paper we show the following conjecture of Noga Alon. Fix a positive integer d>2 and real epsilon > 0; consider the probability that a random d-regular graph on n vertices has the second eigenvalue of its adjacency matrix greater…

Given a rank 3 real arrangement $\mathcal A$ of $n$ lines in the projective plane, the Dirac-Motzkin conjecture (proved by Green and Tao in 2013) states that for $n$ sufficiently large, the number of simple intersection points of $\mathcal…

The harmonic numbers $H_n=\sum_{0<k\ls n}1/k\ (n=0,1,2,\ldots)$ play important roles in mathematics. With helps of some combinatorial identities, we establish the following two congruences:…

In this paper, we prove a number of results providing either necessary or sufficient conditions guaranteeing that the number of real roots of real polynomials of a given degree is either less or greater than a given number. We also provide…

Motivated by a partition inequality of Bessenrodt and Ono, we obtain analogous inequalities for $k$-colored partition functions $p_{-k}(n)$ for all $k\geq2$. This enables us to extend the $k$-colored partition function multiplicatively to a…

Given a partition $\lambda$, we write $e_j(\lambda)$ for the $j^{\textrm{th}}$ elementary symmetric polynomial $e_j$ evaluated at the parts of $\lambda$ and $e_jp_A(n)$ for the sum of $e_j(\lambda)$ as $\lambda$ ranges over the set of…

We resolve the Ramsey problem for $\{x,y,z:x+y=p(z)\}$ for all polynomials $p$ over $\mathbb{Z}$. In particular, we characterise all polynomials that are $2$-Ramsey, that is, those $p(z)$ such that any $2$-colouring of $\mathbb{N}$ contains…

In [1], we give Dickson's conjecture on $N^n$. In this paper, we further give Dickson's conjecture on $Z^n$ and obtain an equivalent form of Green-Tao's conjecture [2]. Based on our work, it is possible to establish a general theory that…

Following Cayley, MacMahon, and Sylvester, define a non-unitary partition to be an integer partition with no part equal to one, and let $\nu(n)$ denote the number of non-unitary partitions of size $n$. In a 2021 paper, the sixth author…

Recently, Andrews and EI Bachraoui obtained several iden tities on two-colored partitions. While solving open problems they posed, Chen and Zhou derived a number of identities using analytic methods and asked for combinatorial proofs. In…

In recent years, the log-concavity of $\{\sqrt[n]{S_n}\}_{n\geq 1}$ have been received a lot of attention. Very recently, Sun posed the following conjecture in his new book: the sequences $\{\sqrt[n]{a_n}\}_{n\geq 2}$ and $\{…

We prove two determinant evaluations attached to Sun's conjectures on matrices of Legendre symbols. The first one resolves the \(p\equiv1\pmod4\) part of Conjecture 4.8(i) by reducing the determinant with four indeterminates to a four-entry…

A partition n = p_1 + p_2 + ... + p_k with 1 <= p_1 <= p_2 <= ... <= p_k is called non-squashing if p_1 + ... + p_j <= p_{j+1} for 1 <= j <= k-1. Hirschhorn and Sellers showed that the number of non-squashing partitions of n is equal to the…

Raimi's theorem guarantees the existence of a partition of $\mathbb{N}$ into two parts with an unavoidable intersection property: for any finite coloring of $\mathbb{N}$, some color class intersects both parts infinitely many times, after…