Related papers: Block partitions: an extended view

In this paper, we are interested in minimizing the sum of block sizes in a pairwise balanced design, where there are some constraints on the size of one block or the size of the largest block. For every positive integers n;m, where m ? n,…

For positive integers $n, L$ and $s$, consider the following two sets that both contain partitions of $n$ with the difference between the largest and smallest parts bounded by $L$: the first set contains partitions with smallest part $s$,…

Let $A_i$ and $B_i$ be positive definite matrices for every $i=1,\cdots,m.$ Let $Z=[Z_{ij}]$ be the block matrix, where $Z_{ij}=B_i^{^\frac{1}{_2}}\left(\displaystyle\sum_{k=1}^mA_k\right)B_j^{^\frac{1}{_2}}$ for every $ i,j=~1,\cdots,m$.…

A partition of degree $n$ is a decomposition $n=i_1+i_2+\dots+i_q$, where ${i_1,i_2,\dots,i_q}$ are positive integers called the parts of the partition. Let $\lambda>0$ be an integer. The partition is said to be a $\lambda$--partition if…

In this paper, we introduce a natural geometric extension of the partition function. More precisely, we investigate the problem of counting partitions of a rectangle into rectangular blocks with integer sides. Here, two partitions of a…

To partition a sequence of n integers into subsets with prescribed sums is an NP-hard problem in general. In this paper we present an efficient solution for the homogeneous version of this problem; i.e. where the elements in each subset add…

If $s$ is a positive integer and $A$ is a set of positive integers, we say that $B$ is an $s$-divisor of $A$ if $\sum_{b\in B} b\mid s\sum_{a\in A} a$. We study the maximal number of $k$-subsets of an $n$-element set that can be…

It is known that for an arbitrary positive integer \(n\) the sequence \(S(x^n)=(1^n, 2^n, \ldots)\) is complete, meaning that every sufficiently large integer is a sum of distinct \(n\)th powers of positive integers. We prove that every…

In recent work, M. Schneider and the first author studied a curious class of integer partitions called "sequentially congruent" partitions: the $m$th part is congruent to the $(m+1)$th part modulo $m$, with the smallest part congruent to…

Given a sequence $s=(s_1,s_2,\ldots)$ of positive integers, the inversion sequences with respect to $s$, or $s$-inversion sequences, were introduced by Savage and Schuster in their study of lecture hall polytopes. A sequence…

The distribution of a given sequence in the set of all sequences with n ones and m = M - n zeros are found by relating the problem to the partitions of a natural number in m natural summands, taking into account the order. The formulas…

In this paper, we investigate the combinatorial properties of three classes of integer partitions: (1) $s$-modular partitions, a class consisting of partitions into parts with a number of occurrences (i.e., multiplicity) congruent to $0$ or…

We introduce the notion of arithmetic progression blocks or AP-blocks of $\mathbb{Z}_n$, which can be represented as sequences of the form $(x, x+m, x+2m, ..., x+(i-1)m) \pmod n$. Then we consider the problem of partitioning $\mathbb{Z}_n$…

We study the rate of growth of $p(n,S,M)$, the number of partitions of $n$ whose parts all belong to $S$ and whose multiplicities all belong to $M$, where $S$ (resp. $M$) are given infinite sets of positive (resp. nonnegative) integers. We…

A set partition of $[n] := \{1, 2, \dots, n \}$ is called {\em $r$-Stirling} if the numbers $1, 2, \dots, r$ belong to distinct blocks. Haglund, Rhoades, and Shimozono constructed graded ring $R_{n,k}$ depending on two positive integers $k…

Let $S$ be the numerical semigroup generated by three consecutive numbers $a,a+1,a+2$, where $a\in\mathbb{N}$, $a\geq 3$. We describe the elements of $S$ whose factorizations have all the same length, as well as the set of factorizations of…

Let $S = \{q_1, \ldots , q_s\}$ be a finite, non-empty set of distinct prime numbers. For a non-zero integer $m$, write $m = q_1^{r_1} \ldots q_s^{r_s} M$, where $r_1, \ldots , r_s$ are non-negative integers and $M$ is an integer relatively…

We exhibit, for any positive integer parameter $s$, an involution on the set of integer partitions of $n$. These involutions show the joint symmetry of the distributions of the following two statistics. The first counts the number of parts…

Let $G$ be a graph(directed or undirected) having $k$ number of blocks. A $\mathcal{B}$-partition of $G$ is a partition into $k$ vertex-disjoint subgraph $(\hat{B_1},\hat{B_1},\hdots,\hat{B_k})$ such that $\hat{B}_i$ is induced subgraph of…

In this paper, we study the "sum composition problem" between two lists $A$ and $B$ of positive integers. We start by saying that $B$ is "sum composition" of $A$ when there exists an ordered $m$-partition $[A_1,\ldots,A_m]$ of $A$ where $m$…