Related papers: On Schreier-type Sets, Partitions, and Composition…

A partition of the positive integers into sets $A$ and $B$ {\em avoids} a set $S\subset\N$ if no two distinct elements in the same part have a sum in $S$. If the partition is unique, $S$ is {\em uniquely avoidable.} For any irrational…

For a numerical semigroup $S := \langle n_1, \dots, n_k \rangle$ with minimal generators $n_1 < \cdots < n_k$, Barron, O'Neill, and Pelayo showed that $L(s+n_1) = L(s) + 1$ and $\ell(s+n_k) = \ell(s) + 1$ for all sufficiently large $s \in…

An indecomposable decomposition of a torsion-free abelian group $G$ of rank $n$ is a decomposition $G=A_1\oplus\cdots\oplus A_t$ where $A_i$ is indecomposable of rank $r_i$ so that $\sum_i r_i=n$ is a partition of $n$. The group $G$ may…

A set $S\subseteq\{0,1\}^E$ of binary vectors, with positions indexed by $E$, is said to be a \textit{powerful code} if, for all $X\subseteq E$, the number of vectors in $S$ that are zero in the positions indexed by $X$ is a power of 2. By…

Let $P$ be the set of integer partitions and $D$ the subset of those with distinct parts. We extend a correspondence of Burge between partitions and binary words to give encodings of both $D$ and $D$ as words over a $k$-ary alphabet, for…

We derive new formulas for the number of unordered (distinct) factorizations with $k$ parts of a positive integer $n$ as sums over the partitions of $k$ and an auxiliary function, the number of partitions of the prime exponents of $n$,…

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…

$\textit{Partial words}$ are words that contain, in addition to letters, special symbols $\diamondsuit$ called $\textit{holes}$. Two partial words of $a=a_0 \dots a_n$ and $b=b_0 \dots b_n$ are $\textit{compatible}$ if for all $i$, $a_i =…

In a recent paper on a study of the Sylow 2-subgroups of the symmetric group with 2^n elements it has been show that the growth of the first (n-2) consecutive indices of a certain normalizer chain is linked to the sequence of partitions of…

For a set of positive integers $A$, let $p_A(n)$ denote the number of ways to write $n$ as a sum of integers from $A$, and let $p(n)$ denote the usual partition function. In the early 40s, Erd\H{o}s extended the classical Hardy--Ramanujan…

Given a sequence of $N$ positive real numbers $\{a_1,a_2,..., a_N \}$, the number partitioning problem consists of partitioning them into two sets such that the absolute value of the difference of the sums of $a_j$ over the two sets is…

Euler showed that the number of partitions of $n$ into distinct parts equals the number of partitions of $n$ into odd parts. This theorem was generalized by Glaisher and further by Franklin. Recently, Beck made three conjectures on…

The partition algebra is an associative algebra with a basis of set-partition diagrams and multiplication given by diagram concatenation. It contains as subalgebras a large class of diagram algebras including the Brauer, planar partition,…

We introduce a family of univariate polynomials indexed by integer partitions. At prime powers, they count the number of subspaces in a finite vector space that transform under a regular diagonal matrix in a specified manner. This…

A finite subset of the natural numbers is weak-Schreier if $\min S \ge |S|$, strong-Schreier if $\min S>|S|$, and maximal if $\min S = |S|$. Let $M_n$ be the number of weak-Schreier sets with $n$ being the largest element and $(F_n)_{n\geq…

A $k$-configuration is a collection of $k$ distinct integers $x_1,\ldots,x_k$ together with their pairwise arithmetic means $\frac{x_i+x_j}{2}$ for $1 \leq i < j \leq k$. Building on recent work of Filmus, Hatami, Hosseini and Kelman on…

We investigate various sparse sets that satisfy the prime number theorem. The sparsest of these sets, $\{\lfloor x/n^t \rfloor:n \le x\}$, has density approaching $1/x$ as $t$ approaches infinity.

Let $G$ be an abelian group of finite order $n$, and let $h$ be a positive integer. A subset $A$ of $G$ is called {\em weakly $h$-incomplete}, if not every element of $G$ can be written as the sum of $h$ distinct elements of $A$; in…

Let (G, *) be a semigroup, D subset of G, and n >= 2 be an integer. We say that (D, *) is an n-closed subset of G if a_1* ... *a_n in D for every a_1, ..., a_n in D. Hence every closed set is a 2-closed set. The concept of n-closed sets…

An integer packing set is a set of non-negative integer vectors with the property that, if a vector $x$ is in the set, then every non-negative integer vector $y$ with $y \leq x$ is in the set as well. Integer packing sets appear naturally…