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Partition and composition calculator; Weisstein, Eric W. "Partition". MathWorld. Wilf, Herbert S. Lectures on Integer Partitions (PDF), archived (PDF) from the original on 2021-02-24; Counting with partitions with reference tables to the On-Line Encyclopedia of Integer Sequences; Integer partitions entry in the FindStat database
The values (), …, of the partition function (1, 2, 3, 5, 7, 11, 15, and 22) can be determined by counting the Young diagrams for the partitions of the numbers from 1 to 8. In number theory, the partition function p(n) represents the number of possible partitions of a non-negative integer n.
However, the coefficient of x 12 is −1 because there are seven ways to partition 12 into an even number of distinct parts, but there are eight ways to partition 12 into an odd number of distinct parts, and 7 − 8 = −1. This interpretation leads to a proof of the identity by canceling pairs of matched terms (involution method). [1]
In the number theory of integer partitions, the numbers () denote both the number of partitions of into exactly parts (that is, sums of positive integers that add to ), and the number of partitions of into parts of maximum size exactly .
Equal-cardinality partition is a variant in which both parts should have an equal number of items, in addition to having an equal sum. This variant is NP-hard too. [5]: SP12 Proof. Given a standard Partition instance with some n numbers, construct an Equal-Cardinality-Partition instance by adding n zeros. Clearly, the new instance has an equal ...
In mathematics, a composition of an integer n is a way of writing n as the sum of a sequence of (strictly) positive integers. Two sequences that differ in the order of their terms define different compositions of their sum, while they are considered to define the same integer partition of that number. Every integer has finitely many distinct ...
In number theory, Glaisher's theorem is an identity useful to the study of integer partitions. Proved in 1883 [ 1 ] by James Whitbread Lee Glaisher , it states that the number of partitions of an integer n {\displaystyle n} into parts not divisible by d {\displaystyle d} is equal to the number of partitions in which no part is repeated d ...
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