منابع مشابه
The Combinatorics of Weighted Vector Compositions
A vector composition of a vector l is a matrix A whose rows sum to l. We define a weighted vector composition as a vector composition in which the column values of A may appear in different colors. We study vector compositions from different viewpoints: (1) We show how they are related to sums of random vectors and (2) how they allow to derive formulas for partial derivatives of composite funct...
متن کاملAn Inequality between Compositions of Weighted Arithmetic and Geometric Means
Let P denote the collection of positive sequences defined on N. Fix w ∈ P. Let s, t, respectively, be the sequences of partial sums of the infinite series ∑ wk and ∑ sk, respectively. Given x ∈ P, define the sequences A(x) and G(x) of weighted arithmetic and geometric means of x by An(x) = n ∑ k=1 wk sn xk, Gn(x) = n ∏ k=1 x wk/sn k , n = 1, 2, . . . Under the assumption that log t is concave, ...
متن کاملSome Elementary Congruences for the Number of Weighted Integer Compositions
An integer composition of a nonnegative integer n is a tuple (π1, . . . , πk) of nonnegative integers whose sum is n; the πi’s are called the parts of the composition. For fixed number k of parts, the number of f -weighted integer compositions (also called f -colored integer compositions in the literature), in which each part size s may occur in f(s) different colors, is given by the extended b...
متن کاملRestricted Weighted Integer Compositions and Extended Binomial Coefficients
We prove a simple relationship between extended binomial coefficients — natural extensions of the well-known binomial coefficients — and weighted restricted integer compositions. Moreover, we give a very useful interpretation of extended binomial coefficients as representing distributions of sums of independent discrete random variables. We apply our results, e.g., to determine the distribution...
متن کاملCompositions versus Cyclic Compositions
We prove that the sum of greatest common divisors of parts in all compositions of n equals the sum of lengths of all cyclic compositions of n. The proof highlights structural similarities between the set of compositions of n and the set of cyclic compositions of n. 1. Definitions and Statement A composition of the positive integer n is an ordered partition of n; that is, a vector ( ) r p p ...,...
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ژورنال
عنوان ژورنال: Canadian Mathematical Bulletin
سال: 1961
ISSN: 0008-4395,1496-4287
DOI: 10.4153/cmb-1961-006-0