Related papers: Cores of partitions in rectangles

An internal partition of a graph is a partitioning of the vertex set into two parts such that for every vertex, at least half of its neighbors are on its side. We prove that for every positive integer $r$, asymptotically almost every…

Partitions, the partition function $p(n)$, and the hook lengths of their Ferrers-Young diagrams are important objects in combinatorics, number theory and representation theory. For positive integers $n$ and $t$, we study $p_t^e(n)$ (resp.…

A partition of $n$ is called a $t$-core partition if none of its hook number is divisible by $t.$ In 2019, Hirschhorn and Sellers \cite{Hirs2019} obtained a parity result for $3$-core partition function $a_3(n)$. Recently, both authors…

We investigate the number of parts modulo $m$ of $m$-ary partitions of a positive integer $n$. We prove that the number of parts is equidistributed modulo $m$ on a special subset of $m$-ary partitions. As consequences, we explain when the…

A partition is $t$-regular if none of its parts is divisible by $t$. Let $p(N,t)$ be the number of $(t+1)$-regular partitions of a positive integer $N$. In 1971, Hagis proved an asymptotic formula for $p(N,t)$ using the circle method, when…

Simultaneous bar-cores, core shifted Young diagrams (or CSYDs), and doubled distinct cores have been studied since Morris and Yaseen introduced the concept of bar-cores. In this paper, our goal is to give a formula for the number of these…

In this paper, we study the $n$-point function of $t$-core partitions. The main tool is the topological vertex, originally developed to study the topological string theory for toric Calabi--Yau 3-folds. By virtue of the topological vertex,…

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…

Simultaneous core partitions have been widely studied in the past 20 years. In 2013, Amdeberhan gave several conjectures on the number, the average size, and the largest size of $(t,t+1)$-core partitions with distinct parts, which was…

Let $p$ be a large prime, $\ell\geq 2$ be a positive integer, $m\geq 2$ be an integer relatively prime to $\ell$ and $P(x)\in\mathbb{F}_p[x]$ be a polynomial which is not a complete $\ell'$-th power for any $\ell'$ for which…

Suppose $s$ and $t$ are coprime positive integers, and let $\sigma$ be an $s$-core partition and $\tau$ a $t$-core partition. In this paper we consider the set $\mathcal P_{\sigma,\tau}(n)$ of partitions of $n$ with $s$-core $\sigma$ and…

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 study the asymptotics of certain measures on partitions (the so-called z-measures and their relatives) in two different regimes: near the diagonal of the corresponding Young diagram and in the intermediate zone between the diagonal and…

In this article we study the "norm" of an integer partition, which we define to be the product of the parts. This partition-theoretic statistic has appeared here and there in the literature of the last century or so, and is at the heart of…

Anderson established a connection between core partitions and order ideals of certain posets by mapping a partition to its $\beta$-set. In this paper, we give a characterization of the poset $P_{(s,s+1,s+2)}$ whose order ideals correspond…

Let r_j(\pi,s) denote the number of cells, colored j, in the s-residue diagram of partition \pi. The GBG-rank of \pi mod s is defined as r_0+r_1*w_s+r_2*w_s^2+...+r_(s-1)*w_s^(s-1), where w_s=exp(2*\Pi*I/s). We will prove that for (s,t)=1,…

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…

For a positive integer $t$, let $b_{t}(n)$ denote the number of $t$-regular partitions of a nonnegative integer $n$. In a recent paper, Keith and Zanello established infinite families of congruences and self-similarity results modulo $2$…

The mass distributions of dense cores in star-forming regions are measured to have a shape similar to the initial mass function of stars. This has been generally interpreted to mean that the constituent cores will form individual stars or…

For a given permutation or set partition there is a natural way to assign a genus. Counting all permutations or partitions of a fixed genus according to cycle lengths or block sizes, respectively, is the main content of this article. After…