Related papers: On the integer partitions recursive structure

In this article, we discuss the notion of partition of elements in an arbitrary Coxeter system $(W,S)$: a partition of an element $w$ is a subset $\mathcal P\subseteq W$ such that the left inversion set of $w$ is the disjoint union of the…

In this note, we consider ordered partitions of integers such that each entry is no more than a fixed portion of the sum. We give a method for constructing all such compositions as well as both an explicit formula and a generating function…

We show that the compositions of positive integers may be interpreted in terms of powers of some power series, over arbitrary commutative ring. As consequences, several closed formulas for the compositions as well as for the generalized…

We show that integer partitions, the fundamental building blocks in additive number theory, detect prime numbers in an unexpected way. Answering a question of Schneider, we show that the primes are the solutions to special equations in…

By using techniques of poset representation theory, we present a formula for the number of partitions of a positive integer into three polygonal numbers.

The aim of this paper is twofold. First, we study the number of partitions of a positive integer $m$ into at most $n$ parts in a given set $A$. We prove that such a number is bounded by the $n$-th Fibonacci number $F(n)$ for any $m$ and…

Cyclotomic polynomials are basic objects in Number Theory. Their properties depend on the number of distinct primes that intervene in the factorization of their order, and the binary case is thus the first nontrivial case. This paper sees…

We use the Circle Method to derive asymptotic formulas for functions related to the number of parts of partitions in particular residue classes.

Given a sequence A=(a1,...,an) of real numbers, a block B of the A is either a set B={ai,...,aj} where i<=j or the empty set. The size b of a block B is the sum of its elements. We show that when 0<=ai<=1 and k is a positive integer, there…

Fibonacci numbers can be expressed in terms of multinomial coefficients as sums over integer partitions into odd parts. We use this fact to introduce a family of double inequalities involving the generating function for the number of…

Say that mu is a ``subpartition'' of an integer partition lambda if the multiset of parts of mu is a submultiset of the parts of lambda, and define an integer partition lambda to be ``wide'' if for every subpartition mu of lambda, mu >= mu'…

A partition of a positive integer $n$ is defined as a non-increasing sequence $P = [y_0, y_1, ..., y_m]$ of positive integers which sum to $n$, where the $y_i$ are called the $parts$ of the partition. A Young diagram is a visual…

If s and t are relatively prime positive integers we show that the s-core of a t-core partition is again a t-core partition

We investigate the representation of symmetric polynomials as a sum of squares. Since this task is solved using semidefinite programming tools we explore the geometric, algebraic, and computational implications of the presence of discrete…

The infinite models in integer programming can be described as the convex hull of some points or as the intersection of halfspaces derived from valid functions. In this paper we study the relationships between these two descriptions. Our…

A universal generator for integer-valued square-integrable random variables is introduced. The generator relies on a rejection technique based on a generalization of the inversion formula for integer-valued random variables. The proposal…

For integer $g\ge 5$, we prove that any positive integer can be written as a sum of three palindromes in base $g$.

In his classic text, \emph{Combinatory Analysis}, MacMahon defined a perfect partition of a positive integer $n$ as a partition whose parts contain exactly one partition of every positive integer not exceeding $n$. In this paper we apply…

A positive integer is expressed as a sum of squares of positive integers in a unique way applying a special technique. The expression, thus obtained is resolved into two factors using the concept of the Clifford algebra. This technique is…

The representation of any integer as the sum of two cubes to a fixed modulus is always possible if and only if the modulus is not divisible by seven or nine. For a positive non-prime integer N there is given an inductive way to find its…