Partition (number theory)

About: Partition (number theory) is a research topic. Over the lifetime, 3580 publications have been published within this topic receiving 47107 citations. The topic is also known as: integer partition & Partition (number theory).

The theory of partitions

Abstract: 1. The elementary theory of partitions 2. Infinite series generating functions 3. Restricted partitions and permutations 4. Compositions and Simon Newcomb's problem 5. The Hardy-Ramanujan-Rademacher expansion of p(n) 6. The asymptotics of infinite product generating functions 7. Identities of the Rogers-Ramanujan type 8. A general theory of partition identities 9. Sieve methods related to partitions 10. Congruence properties of partition functions 11. Higher-dimensional partitions 12. Vector or multipartite partitions 13. Partitions in combinatorics 14. Computations for partitions.

A New Vector Partition of the Probability Score

Abstract: A new vector partition of the probability, or Brier, score (PS) is formulated and the nature and properties of this partition are described. The relationships between the terms in this partition and the terms in the original vector partition of the PS are indicated. The new partition consists of three terms: 1) a measure of the uncertainty inherent in the events, or states, on the occasions of concern (namely, the PS for the sample relative frequencies); 2) a measure of the reliability of the forecasts; and 3) a new measure of the resolution of the forecasts. These measures of reliability and resolution are and are not, respectively, equivalent (i.e., linearly related) to the measures of reliability and resolution provided by the original partition. Two sample collections of probability forecasts are used to illustrate the differences and relationships between these partitions. Finally, the two partitions are compared, with particular reference to the attributes of the forecasts with which the pa...

Algorithm 448: number of multiply-restricted partitions

Abstract: Given a positiver integer m and an ordered k-tuple c = (c1, ··· , ck) of not necessarily distinct positive integers, then any ordered k-tuple s = (s1, ··· , sk) of nonnegative integers such that m = ∑ki-1sici is said to be a partition of m restricted to c. Let Pc(m) denote the number of distinct partitions of m restricted to c. The subroutine COUNT, when given a k-tuple c and an integer n, computes an array of the values of Pc(m) for m = 1 to n. Many combinatorial enumeration problems may be expressed in terms of the numbers Pc(m). We mention two below.

Exchangeable and partially exchangeable random partitions

Abstract: Call a random partition of the positive integerspartially exchangeable if for each finite sequence of positive integersn 1,...,n k, the probability that the partition breaks the firstn 1+...+nk integers intok particular classes, of sizesn 1,...,nk in order of their first elements, has the same valuep(n 1,...,nk) for every possible choice of classes subject to the sizes constraint. A random partition is exchangeable iff it is partially exchangeable for a symmetric functionp(n 1,...nk). A representation is given for partially exchangeable random partitions which provides a useful variation of Kingman's representation in the exchangeable case. Results are illustrated by the two-parameter generalization of Ewens' partition structure.