Related papers: Armstrong's Conjecture for $(k, mk + 1)$-Core Part…

A partition is a $t$-core partition if $t$ is not one of its hook lengths. Let $c_t(N)$ be the number of $t$-core partitions of $N$. In 1999, Stanton conjectured $c_t(N) \le c_{t+1}(N)$ if $4 \le t \ne N-1$. This was proved for $t$ fixed…

We prove that the number of partitions of an integer into at most b distinct parts of size at most n forms a unimodal sequence for n sufficiently large with respect to b. This resolves a recent conjecture of Stanley and Zanello.

In this paper, we study $(s,s+1)$-core partitions with $d$-distinct parts. We obtain results on the number and the largest size of such partitions, so we extend Xiong's paper in which the results are obtained about $(s,s+1)$-core partitions…

Fix $t \geq 2$. We first give an asymptotic formula for certain sums of the number of $t$-cores. We then use this result to compute the distribution of the size of the $t$-core of a uniformly random partition of an integer $n$. We show that…

We present a conjecture about partitions, with a very elementary formulation.

Integer partitions which are simultaneously $t$--cores for distinct values of $t$ have attracted significant interest in recent years. When $s$ and $t$ are relatively prime, Olsson and Stanton have determined the size of the maximal…

The ABC conjecture of Masser and Oesterle' states that if (a,b,c) are coprime integers with a + b + c = 0, then sup(|a|,|b|,|c|) < c_e (rad(abc))^{1+e} for any e > 0. Oesterle' has observed that if the ABC conjecture holds for all (a,b,c)…

A theorem of Andrews equates partitions in which no part is repeated more than 2k-1 times to partitions in which, if j appears at least k times, all parts less than j also do so. This paper proves the theorem bijectively, with some of the…

Amdeberhan conjectured that the number of $(s,s+2)$-core partitions with distinct parts for an odd integer $s$ is $2^{s-1}$. This conjecture was first proved by Yan, Qin, Jin and Zhou, then subsequently by Zaleski and Zeilberger. Since the…

Integer partitions have long been of interest to number theorists, perhaps most notably Ramanujan, and are related to many areas of mathematics including combinatorics, modular forms, representation theory, analysis, and mathematical…

The Standard Simplex Conjecture of Isaksson and Mossel asks for the partition $\{A_{i}\}_{i=1}^{k}$ of $\mathbb{R}^{n}$ into $k\leq n+1$ pieces of equal Gaussian measure of optimal noise stability. That is, for $\rho>0$, we maximize $$…

We prove three main conjectures of Berkovich and Uncu (Ann. Comb. 23 (2019) 263--284) on the inequalities between the numbers of partitions of $n$ with bounded gap between largest and smallest parts for sufficiently large $n$. Actually our…

Let $b_{t,i}(n)$ denote the total number of the $i$ hooks in the $t$-regular partitions of $n$. Singh and Barman (J. Number Theory { 264} (2024), 41--58) raised two conjectures on $b_{t,i}(n)$. The first conjecture is on the positivity of…

If $a_1, a_2, ..., a_k$ and $n$ are positive integers such that $n = a_1 + a_2 + ... + a_k$, then the sum $a_1 + a_2 + ... + a_k$ is said to be a \emph{partition of $n$} of \emph{length $k$}, and $a_1, a_2, ..., a_k$ are said to be the…

Given three positive integers $n,a,b$ with $n=ab$, we determine the base size of the symmetric group and of the alternating group of degree $n$ in their action on the set of partitions into $b$ parts having cardinality $a$.

In 2007, Olsson and Stanton gave an explicit form for the largest $(a, b)$-core partition, for any relatively prime positive integers $a$ and $b$, and asked whether there exists an $(a, b)$-core that contains all other $(a, b)$-cores as…

A scarcely known generalization of Goldbach's conjecture introduced by Hardy and Littlewood states that for every pair of (relatively prime) positive integers m1 and m2, every sufficiently large integer n satisfying certain simple…

We conjecture that every infinite group $G$ can be partitioned into countably many cells $G=\bigcup_{n\in\omega}A_n$ such that $cov(A_nA_n^{-1})=|G|$ for each $n\in\omega$. Here $cov(A)=\min\{|X|:X\subseteq G, G=XA\}$. We confirm this…

For fixed s, the size of an (s, s+1)-core partition with distinct parts can be seen as a random variable X_s. Using computer-assisted methods, we derive formulas for the expectation, variance, and higher moments of X_s. Our results give…

Simultaneous core partitions are important objects in algebraic combinatorics. Recently there has been interest in studying the distribution of sizes among all $(s,t)$-cores for coprime $s$ and $t$. Zaleski (2017) gave strong evidence that…