Related papers: Integer compositions with part sizes not exceeding…
A well-known result from Hardy and Ramanujan gives an asymptotic expression for the number of possible ways to express an integer as the sum of smaller integers. In this vein, we consider the general partitioning problem of writing an…
The concept of graph compositions is related to several number theoretic concepts, including partitions of positive integers and the cardinality of the power set of finite sets. This paper examines graph compositions where the total number…
We investigate some types of composition operators, linear and not, and conditions for some spaces to be mapped into themselves and for the operators to satisfy some good properties.
Automatic chess problem or puzzle composition typically involves generating and testing various different positions, sometimes using particular piece sets. Once a position has been generated, it is then usually tested for positional…
We enumerate permutations that avoid all but one of the $k$ patterns of length $k$ starting with a monotone increasing subsequence of length $k-1$. We compare the size of such permutation classes to the size of the class of permutations…
We carry out the asymptotic analysis of repulsive ensembles of N particles which are discrete analogues of continuous 1d log-gases or beta-ensembles of random matrix theory. The ensembles that we study have several groups of particles which…
Let $k\ge 1$ be an integer. We prove that a suitable asymptotic formula for the average number of representations of integers $n=p_{1}^{k}+p_{2}^{2}+p_{3}^{2}$, where $p_1,p_2,p_3$ are prime numbers, holds in intervals shorter than the ones…
The study of pattern avoidance in permutations, and specifically in flattened partitions is an active area of current research. In this paper, we count the number of distinct flattened partitions over [n] avoiding a single pattern, as well…
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…
We investigate the problem of finding integers $k$ such that appending any number of copies of the base-ten digit $d$ to $k$ yields a composite number. In particular, we prove that there exist infinitely many integers coprime to all digits…
In this article, we introduce the notion of almost consecutive partitions. A partition is almost consecutive if every term is consecutive, with the possible exception of the smallest one. We find formulas relating to the smallest parts of…
We consider the problem of reconstructing compositions of an integer from their subcompositions, which was raised by Raykova (albeit disguised as a question about layered permutations). We show that every composition w of n\ge 3k+1 can be…
In this note, I discuss results on integer compositions/partitions given in the paper "A Unified Approach to Algorithms Generating Unrestricted and Restricted Integer Compositions and Integer Partitions". I also experiment with four…
In the classical non-adaptive group testing setup, pools of items are tested together, and the main goal of a recovery algorithm is to identify the "complete defective set" given the outcomes of different group tests. In contrast, the main…
In this undergraduate thesis, we expand on the study of statistics on restricted growth functions avoiding patterns initiated by Campbell, et. al. Restricted growth functions are of interest because they are in bijection with set…
We compute the lengths of two particular cases of (possibly non-unital) composition algebras, namely, standard composition algebras and Okubo algebras over an arbitrary field $\mathbb{F}$. These results finish the complete description of…
The investigation of colour symmetries for periodic and aperiodic systems consists of two steps. The first concerns the computation of the possible numbers of colours and is mainly combinatorial in nature. The second is algebraic and…
Analytic combinatorics studies the asymptotic behaviour of sequences through the analytic properties of their generating functions. This article provides effective algorithms required for the study of analytic combinatorics in several…
Following Cayley, MacMahon, and Sylvester, define a non-unitary partition to be an integer partition with no part equal to one, and let $\nu(n)$ denote the number of non-unitary partitions of size $n$. In a 2021 paper, the sixth author…
This article investigates integer sequences that partition the sequence into blocks of various lengths - irregular arrays. The main result of the article is explicit formulas for numbering of irregular arrays. A generalization of Cantor…