相关论文: Generalized compositions with a fixed number of pa…
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…
In this paper we consider particular generalized compositions of a natural number with a given number of parts. Its number is a weighted polynomial coefficient. The number of all generalized compositions of a natural number is a weighted…
A generalization of the Catalan numbers is considered. New results include binomial identities, recursive relations and a close formula for the multivariate generating function. A simple expression for the Catalan determinant is derived.
We consider two type of upper Hessenberg matrices which determinants are Fibonacci numbers. Calculating sums of principal minors of the fixed order of the first type leads us to convolved Fibonacci numbers. Some identities for these and for…
A new class of alternating convolutions concerning binomial coefficients and Catalan numbers are evaluated in closed forms.
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…
A composition of a nonnegative integer (n) is a sequence of positive integers whose sum is (n). A composition is palindromic if it is unchanged when its terms are read in reverse order. We provide a generating function for the number of…
We first give a combinatorial interpretation of coefficients of Chebyshev polynomials, which allows us to connect them with compositions of natural numbers. Then we describe a relationship between the number of compositions of a natural…
This work is a study of polynomial compositions having a fixed number of terms. We outline a recursive method to describe these characterizations, give some particular results and discuss the general case. In the final sections, some…
We define a new generalization of Catalan numbers to multinomial coefficients. With arithmetic methods, we study their integrality and the integrality of their Lucasnomial generalization. We give a complete characterization of regular Lucas…
We show that the compositions of positive integers may be interpreted in terms of powers of some power series, over arbitrary commutative ring. As consequences, several closed formulas for the compositions as well as for the generalized…
A composition of $n\in\NN$ is an ordered collection of one or more positive integers whose sum is $n$. The number of summands is called the number of parts of the composition. A palindromic composition of $n$ is a composition of $n$ in…
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…
The Catalan numbers are well-known to be the answer to many different counting problems, and so there are many different families of sets whose cardinalities are the Catalan numbers. We show how such a family can be given the structure of a…
An integer composition of a nonnegative integer $n$ is a tuple $(\pi_1,\ldots,\pi_k)$ of nonnegative integers whose sum is $n$; the $\pi_i$'s are called the parts of the composition. For fixed number $k$ of parts, the number of $f$-weighted…
We consider colored compositions where only some parts are allowed different colors, depending on their locations in the composition. The counting sequences are obtained through generating functions. Connections to many other combinatorial…
We prove astonishing identities generated by compositions of positive integers. In passing, we obtain two new identities for Stirling numbers of the first kind. In the two last sections we clarify an algebraic sense of these identities and…
We define and study the combinatorial properties of compositional Bernoulli numbers and polynomials within the framework of rational combinatorics.
We study the compositions of an integer n whose part sizes do not exceed a fixed integer k. We use the methods of analytic combinatorics to obtain precise asymptotic formulas for the number of such compositions, the total number of parts…
Graph compositions generalize both integer compositions and partitions of a finite set. We develop formulas, generating functions and recurrence relations for composition counting functions for several families of graphs.