Related papers: Compositions with restricted parts

Wilf's Sixth Unsolved Problem asks for any interesting properties of the set of partitions of integers for which the (nonzero) multiplicities of the parts are all different. We refer to these as \emph{Wilf partitions}. Using $f(n)$ to…

Recently, Andrews and El Bachraoui considered the number of integer partitions whose smallest part is repeated exactly $k$ times and the remaining parts are not repeated. They presented several interesting results and posed questions…

Let $ped(n)$ denote the number of partitions of $n$ wherein even parts are distinct (and odd parts are unrestricted). We show infinite families of congruences for $ped(n)$ modulo $8$. We also examine the behavior of $ped_{-2}(n)$ modulo $8$…

Let $p_k(n)$ denote the number of $2$-color partitions of $n$ where one of the colors appears only in parts that are multiples of $k$. We will prove a conjecture of Ahmed, Baruah, and Dastidar on congruences modulo $5$ for $p_k(n)$.…

We prove that the number of partitions of an integer into at most b distinct parts of size at most n forms a unimodal sequence for n sufficiently large with respect to b. This resolves a recent conjecture of Stanley and Zanello.

This paper is concerned with structures of general graphs with perfect matchings. We first reveal a partially ordered structure among factor-components of general graphs with perfect matchings. Our second result is a generalization of…

Recently, George Andrews has given a Glaisher style proof of a finite version of Euler's partition identity. We generalise this result by giving a finite version of Glaisher's partition identity. Both the generating function and bijective…

For a class of random partitions of an infinite set a de Finetti-type representation is derived, and in one special case a central limit theorem for the number of blocks is shown.

Motivated by the study of integer partitions, we consider partitions of integers into fractions of a particular form, namely with constant denominators and distinct odd or even numerators. When numerators are odd, the numbers of partitions…

Recently Andrews and Bachraoui proved identities relating certain restricted partitions into distinct even parts with restricted 4-regular partitions by the theory of basic hypergeometric series. They also posed a question regarding…

Hirschhorn and Sellers studied arithmetic properties of the number of partitions with odd parts distinct. In another direction, Hammond and Lewis investigated arithmetic properties of the number of bipartitions. In this paper, we consider…

We show that the discretized configuration space of $k$ points in the $n$-simplex is homotopy equivalent to a wedge of spheres of dimension $n-k+1$. This space is homeomorphic to the order complex of the poset of ordered partial partitions…

Given integers $a_1, a_2, ..., a_n$, with $a_1 + a_2 + ... + a_n \geq 1$, a symmetrically constrained composition $\lambda_1 + lambda_2 + ... + lambda_n = M$ of $M$ into $n$ nonnegative parts is one that satisfies each of the the $n!$…

Two of the pillars of combinatorics are the notion of choosing an arbitrary subset of a set with $n$ elements (which can be done in $2^n$ ways), and the notion of choosing a $k$-element subset of a set with $n$ elements (which can be done…

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…

The celebrated Rogers-Ramanujan identities equate the number of integer partitions of $n$ ($n\in\mathbb N_0$) with parts congruent to $\pm 1 \pmod{5}$ (respectively $\pm 2 \pmod{5}$) and the number of partitions of $n$ with super-distinct…

We obtain congruences for the number a(n) of cubic partitions using modular forms. The notion of cubic partitions is introduced by Chan and named by Kim in connection with Ramanujan's cubic continued fractions. Chan has shown that a(n) has…

We introduce the theory of div point sets, which aims to provide a framework to study the combinatoric nature of any set of points in general position on an Euclidean plane. We then show that proving the unsatisfiability of some first-order…

Let c^{k,l}(n) be the number of compositions (ordered partitions) of the integer n whose Ferrers diagram fits inside a k-by-l rectangle. The purpose of this note is to give a simple, algebraic proof of a conjecture of Vatter that the…

In this paper, we prove a conjecture proposed by George Beck, which involves gap-free partitions and partitions with distinct parts.