Related papers: Unordered Factorizations with $k$ Parts

We visualize the identity p(n) = sum s(k) p(n-k)/n for the integer partition function p(n) involving the divisor function s, add comments on the history of visualizations of numbers, illustrate how different mathematical fields play…

Let $N(n)$ denote the number of isomorphism types of groups of order $n$. We consider the integers $n$ that are products of at most $4$ not necessarily distinct primes and exhibit formulas for $N(n)$ for such $n$.

We introduce and consider a certain probability question involving elementary number theory and the likelihood that a fixed prime will appear in a certain recursively defined factorization of an integer. We derive several convergent…

Euler's theorem asserts that $A(n)=B(n)$ where $A(n)$ is the number of partitions of $n$ into distinct parts and $B(n)$ is the number of partitions of $n$ into odd parts. In this paper, it is proved that for $n>0$, \begin{align*}…

We define a quantity $c_m(n,k)$ as a generalization of the notion of the composition of the positive integer $n$ into $k$ parts. We proceed to derive some known properties of this quantity. In particular, we relate two partial Bell…

In this article, we prove some factorization results for several classes of polynomials having integer coefficients, which in particular yield several classes of irreducible polynomials. Such classes of polynomials are devised by imposing…

For all positive integers $k,l,n$, the Little Glaisher theorem states that the number of partitions of $n$ into parts not divisible by $k$ and occurring less than $l$ times is equal to the number of partitions of $n$ into parts not…

In this paper we give an additive representation of the factorial, which can be proven by a simple quick analytical argument. We also present some generalizations, which are linked, on the one hand to an arithmetical theorem proven by Euler…

We study set partitions with $r$ distinguished elements and block sizes found in an arbitrary index set $S$. The enumeration of these $(S,r)$-partitions leads to the introduction of $(S,r)$-Stirling numbers, an extremely wide-ranging…

We prove a lemma that is useful to get upper bounds for the number of partitions without a given subsum. From this we can deduce an improved upper bound for the number of sets represented by the (unrestricted or into unequal parts)…

We show that, for a certain class of partitions and an even number of variables of which half are reciprocals of the other half, Schur polynomials can be factorized into products of odd and even orthogonal characters. We also obtain related…

Two doubly indexed families of homogeneous and isobaric polynomials in several indeterminates are considered: the (partial) exponential Bell polynomials $B_{n,k}$ and a new family $S_{n,k}$ such that $X_1^{-(2n-1)}S_{n,k}$ and $B_{n,k}$…

Recent developments of Baxter algebras have lead to applications to combinatorics, number theory and mathematical physics. We relate Baxter algebras to Stirling numbers of the first kind and the second kind, partitions and multinomial…

We introduce several statistics on ordered partitions of sets, that is, set partitions where the blocks are permuted arbitrarily. The distribution of these statistics is closely related to the q-Stirling numbers of the second kind. Some of…

In this paper, we will introduce Bell numbers $D(n)$ of type $D$ as an analogue to the classical Bell numbers related to all the partitions of the set $[n]$. Then based on a signed set partition of type $D$, we will construct the recurrence…

In this paper we derive congruences expressing Bell numbers and derangement numbers in terms of each other modulo any prime.

This work is about a partition problem which is an instance of the distance magic graph labeling problem. Given positive integers $n,k$ and $p_1\le p_2\le \cdots\le p_k$ such that $p_1+\cdots+p_k=n$ and $k$ divides $\sum_{i=1}^ni$, we study…

To factor an integer N, given that it is equal to the product of two primes, it suffices to find an integer d satisfying a certain simple numerical test. In this approach, the factorization problem equates to the problem of designing an…

I propose two simple ways of generating the partitions of (n+1) from the partitions of n. A recurrence relation for P(n+1), the number of partitions of (n+1), in terms of P(n) and Q(n), where Q(n) denotes the number of partitions of n…

Let $w$ be a word in alphabet $\{x,D\}$ with $m$ $x$'s and $n$ $D$'s. Interpreting "$x$" as multiplication by $x$, and "$D$" as differentiation with respect to $x$, the identity $wf(x) = x^{m-n}\sum_k S_w(k) x^k D^k f(x)$, valid for any…