Results 1 to 10 of about 904,425 (75)
On the Enumeration and Asymptotic Analysis of Fibonacci Compositions [PDF]
arXiv, 2021 We study Fibonacci compositions, which are compositions of natural numbers that only use Fibonacci numbers, in two different contexts. We first prove inequalities comparing the number of Fibonacci compositions to regular compositions where summands have a maximum possible value.
arxiv
Local Method for Compositional Inverses of Permutational Polynomials [PDF]
arXiv, 2022 In this paper, we provide a local method to find compositional inverses of all PPs, some new PPs and their compositional inverses are given.
arxiv
Anti-palindromic compositions [PDF]
arXiv, 2021 A palindromic composition of $n$ is a composition of $n$ which can be read the same way forwards and backwards. In this paper we define an anti-palindromic composition of $n$ to be a composition of $n$ which has no mirror symmetry amongst its parts. We then give a surprising connection between the number of anti-palindromic compositions of $n$ and the ...
arxiv
On compositions of natural numbers [PDF]
arXiv, 2020 In this expository note, we introduce the reader to compositions of a natural number, e.g., $2+1+2+1+7+1$ is a composition of 14, and $1+2$ and $2+1$ are two different compositions of 3. We discuss some simple restricted forms of compositions, e.g., $23+17+33$ is a composition of 73 into three odd parts.
arxiv
Integer properties of a composition of exponential generating functions [PDF]
arXiv, 2012 In this paper, we study a composition of exponential generating functions. We obtain new properties of this composition, which allow to distinguish prime numbers from composite numbers. Using the result of paper we get the known properties of the Bell numbers(Touchard's Congruence for $k=0$) and new properties of the Euler numbers.
arxiv
Compositions, Partitions, and Fibonacci Numbers [PDF]
Fibonacci Quarterly 40 (2011) 348-354, 2013 A bijective proof is given for the following theorem: the number of compositions of n into odd parts equals the number of compositions of n + 1 into parts greater than one. Some commentary about the history of partitions and compositions is provided.
arxiv
Generalized Compositions of Natural Numbers [PDF]
arXiv, 2010 We consider compositions of natural numbers when there are different types of each natural number. Several recursions as well as some closed formulas for the number of compositions is derived. We also find its relationships with some known classes of integers such as Fibonacci, Catalan, Pell, Pell-Lucas, and Jacobsthal numbers.
arxiv
Generalized compositions with a fixed number of parts [PDF]
arXiv, 2010 We investigate compositions of a positive integer with a fixed number of parts, when there are several types of each natural number. These compositions produce new relationships among binomial coefficients, Catalan numbers, and numbers of the Catalan triangle.
arxiv
Palindromic and Colored Superdiagonal Compositions [PDF]
arXiv, 2021 A superdiagonal composition is one in which the $i$-th part or summand is of size greater than or equal to $i$. In this paper, we study the number of palindromic superdiagonal compositions and colored superdiagonal compositions. In particular, we give generating functions and explicit combinatorial formulas involving binomial coefficients and Stirling ...
arxiv
Composition-theoretic series in partition theory [PDF]
arXiv, 2022 We use sums over integer compositions analogous to generating functions in partition theory, to express certain partition enumeration functions as sums over compositions into parts that are $k$-gonal numbers; our proofs employ Ramanujan's theta functions.
arxiv