Related papers: Finite Differences of the Logarithm of the Partiti…

Let $\bar{p}(n)$ denote the number of overpartitions of $n$. It was conjectured by Hirschhorn and Sellers that $\bar{p}(40n+35)\equiv 0\ ({\rm mod\} 40)$ for $n\geq 0$. Employing 2-dissection formulas of quotients of theta functions due to…

The coefficients of the generating function $(q;q)^\alpha_\infty$ produce $p_\alpha(n)$ for $\alpha \in \mathbb{Q}$. In particular, when $\alpha = -1$, the partition function is obtained. Recently, Chan and Wang identified and proved…

A famous conjecture of Parkin-Shanks predicts that $p(n)$ is odd with density $1/2$. Despite the remarkable amount of work of the last several decades, however, even showing this density is positive seems out of reach. In a 2018 paper with…

In this note, we show that there are many infinity positive integer values of $n$ in which, the following inequality holds $$ \left\lfloor{1/2}(\frac{(n+1)^2}{\log(n+1)}-\frac{n^2}{\log n})-\frac{\log^2 n}{\log\log…

Let $\Sym{n}$ denote the set of all permutations on $n$ labels. Let $c:[0, 1]^2\to [0, \infty)$ be a twice continuously differentiable function. A subfamily of the Mallows model is the Gibbs probability measures on $\Sym{n}$ such that…

Ramanujan's congruence $p(5k+4) \equiv 0 \pmod 5$ led Dyson \cite{dyson} to conjecture the existence of a measure "rank" such that $p(5k+4)$ partitions of $5k+4$ could be divided into sub-classes with equal cardinality to give a direct…

Building on work of Hardy and Ramanujan, Rademacher proved a well-known formula for the values of the ordinary partition function $p(n)$. More recently, Bruinier and Ono obtained an algebraic formula for these values. Here we study the…

We consider infinite sequences of positive numbers. The connection between log-concavity and the Bessenrodt--Ono inequality had been in the focus of several papers. This has applications in the white noise distribution theory and…

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…

If $ p_k(a,m,n) $ denotes the number of partitions of $n$ into $k$th powers with a number of parts that is congruent to $ a $ modulo $m,$ then $p_2(0,2,n)\sim p_2(1,2,n)$ and the sign of the difference $p_2(0,2,n)- p_k(1,2,n)$ alternates…

For every nonnegative integer $n$, let $r_F(n)$ be the number of ways to write $n$ as a sum of Fibonacci numbers, where the order of the summands does not matter. Moreover, for all positive integers $p$ and $N$, let \begin{equation*}…

Let $f \in C^n(\mathbb{R})$ be such that $\Vert f^{(n)} \Vert_\infty < \infty$. Let $f^{[n]} \in C(\mathbb{R}^{n+1})$ be the $n$th order divided difference. A special case of our main result states that for $1 < p < \infty$ we have \[\Vert…

We make an application of ideas from partition theory to a problem in multiplicative number theory. We propose a deterministic model of prime number distribution, from first principles related to properties of integer partitions, that…

We prove new formulas and congruences for $p(n,k):=$ the number of partitions of $n$ into $k$ parts and $q(n,k):=$ the number of partitions of $n$ into $k$ distinct parts. Also, we give lower and upper bounds for the density of the set…

The number partition problem is a well-known problem, which is one of 21 Karp's NP-complete problems \cite{karp}. The partition function is a boolean function that is equivalent to the number partition problem with number range restricted.…

The spt-function spt($n$) was introduced by Andrews as the weighted counting of partitions of $n$ with respect to the number of occurrences of the smallest part. In this survey, we summarize recent developments in the study of spt($n$),…

The parity of the partition function $p(n)$ remains strikingly mysterious. Beyond a handful of fragmentary results, essentially nothing is known about the distribution of parity. We prove a uniform result on quadratic progressions. If…

Let $\Delta_d^{(a,-)}(n) = q_d^{(a)}(n) - Q_d^{(a,-)}(n)$ where $q_d^{(a)}(n)$ counts the number of partitions of $n$ into parts with difference at least $d$ and size at least $a$, and $Q_d^{(a,-)}(n)$ counts the number of partitions into…

Let $k\ge 2$ be a positive integer and $P^+(n)$ the greatest prime factor of a positive integer $n$ with convention $P^+(1)=1$. For any $\theta\in \left[\frac 1{2k},\frac{17}{32k}\right)$, set…

We consider the equal sum partition problem, motivated by distance magic graph labeling: Given $n,k \in \N$ such that $k\, | \sum_{i=1}^ni$ and a partition $p_1+\cdots+p_k=n$, when is it possible to find a partition of the set…