Related papers: The largest singletons of set partitions

Recently, Deutsch and Elizalde studied the largest and the smallest fixed points of permutations. Motivated by their work, we consider the analogous problems in weighted set partitions. Let $A_{n,k}(\mathbf{t})$ denote the total weight of…

Amdeberhan's conjectures on the enumeration, the average size, and the largest size of $(n,n+1)$-core partitions with distinct parts have motivated many research on this topic. Recently, Straub and Nath-Sellers obtained formulas for the…

Let $\mathcal{A}=(a_n)_{n\in\mathbb{N}_+}$ be a sequence of positive integers. Let $p_\mathcal{A}(n,k)$ denote the number of multi-color partitions of $n$ into parts in $\{a_1,\ldots,a_k\}$. We examine several arithmetic properties of the…

An $(n,k)$-Sperner partition system is a set of partitions of some $n$-set such that each partition has $k$ nonempty parts and no part in any partition is a subset of a part in a different partition. The maximum number of partitions in an…

A $k$-regular partition into distinct parts is a partition into distinct parts with no part divisible by $k$. In this paper, we provide a general method to establish the unimodality of $k$-regular partition into distinct parts where the…

In a previous paper (arXiv:1608.02262), we used computer-assisted methods to find explicit expressions for the moments of the size of a uniform random (n,n+1)-core partition with distinct parts. In particular, we conjectured that the…

Let \(\mathcal{P}(n)\) be the set of partitions of the positive integer \(n\). For \(\alpha=(\alpha_1,...,\alpha_t) \in \mathcal{P}(n)\) define the diagonal sequence \(\delta(\alpha)=(d_k(\alpha))_{k \geq 1}\) via \( d_k(\alpha) =…

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…

Let p(n, k) denote the number of partitions of n into parts less than or equal to k. We show several properties of this function modulo 2. First, we prove that for fixed positive integers k and m, p(n,k) is periodic modulo m. Using this, we…

An $(n,k)$-Sperner partition system is a collection of partitions of some $n$-set, each into $k$ nonempty classes, such that no class of any partition is a subset of a class of any other. The maximum number of partitions in an…

We study the maximum multiplicity $\mathcal{M}(k,n)$ of a simple transposition $s_k=(k \: k+1)$ in a reduced word for the longest permutation $w_0=n \: n-1 \: \cdots \: 2 \: 1$, a problem closely related to much previous work on sorting…

Given relatively prime positive integers, $a_1,\ldots,a_n$, the Frobenius number is the largest integer with no representations of the form $a_1x_1+\cdots+a_nx_n$ with nonnegative integers $x_i$. This classical value has recently been…

The (n,k)-arrangement graph A(n,k) is a graph with all the k-permutations of an n-element set as vertices where two k-permutations are adjacent if they agree in exactly k-1 positions. We introduce a cyclic decomposition for k-permutations…

For a prime number $p$ and a sequence of integers $a_0,\dots,a_k\in \{0,1,\dots,p\}$, let $s(a_0,\dots,a_k)$ be the minimum number of $(k+1)$-tuples $(x_0,\dots,x_k)\in A_0\times\dots\times A_k$ with $x_0=x_1+\dots + x_k$, over subsets…

Let $S_{n}$ denote the set of permutations of $[n]=\{1,2,\dots, n\}$. For each integer $k\geq 1$, let $S_{n,k}$ be the set of all permutations of $[n]$ with exactly $k$ disjoint cycles. A subset $H\subseteq S_{n,k}$ is to be a matching if…

A new formula for the partition function $p(n)$ is developed. We show that the number of partitions of $n$ can be expressed as the sum of a simple function of the two largest parts of all partitions. Specifically, if $a_1 + >... + a_k = n$…

The lattice of partitions of a set and its d-divisible generalization have been much studied for their combinatorial, topological, and representation-theoretic properties. An ordered set partition is a set partition where the subsets are…

Let $n, k$ and $a$ be positive integers. The Stirling numbers of the first kind, denoted by $s(n,k)$, count the number of permutations of $n$ elements with $k$ disjoint cycles. Let $p$ be a prime. In recent years, Lengyel, Komatsu and…

In this paper we study combinatorial aspects of permutations of $\{1,\ldots,n\}$ and related topics. In particular, we prove that there is a unique permutation $\pi$ of $\{1,\ldots,n\}$ such that all the numbers $k+\pi(k)$ ($k=1,\ldots,n$)…

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…