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and the number of partitions of n in which all parts are 1, 2 or 3 (or, equivalently, the number of partitions of n into at most three parts) is the nearest integer to (n + 3) 2 / 12. [ 14 ] Partitions in a rectangle and Gaussian binomial coefficients
The function q(n) gives the number of these strict partitions of the given sum n. For example, q(3) = 2 because the partitions 3 and 1 + 2 are strict, while the third partition 1 + 1 + 1 of 3 has repeated parts. The number q(n) is also equal to the number of partitions of n in which only odd summands are permitted. [20]
Generally, a partition is a division of a whole into non-overlapping parts. Among the kinds of partitions considered in mathematics are partition of a set or an ordered partition of a set, partition of a graph, partition of an integer, partition of an interval, partition of unity, partition of a matrix; see block matrix, and
In particular, the left hand side is a generating function for the number of partitions of n into an even number of distinct parts minus the number of partitions of n into an odd number of distinct parts. Each partition of n into an even number of distinct parts contributes +1 to the coefficient of x n; each partition into an odd number of ...
An r-associated Stirling number of the second kind is the number of ways to partition a set of n objects into k subsets, with each subset containing at least r elements. [17] It is denoted by S r ( n , k ) {\displaystyle S_{r}(n,k)} and obeys the recurrence relation
The number of such partitions or equivalence relations is by definition the Stirling number of the second kind S (n, x), also written {}. Its value can be described using a recursion relation or using generating functions , but unlike binomial coefficients there is no closed formula for these numbers that does not involve a summation .
Thus, in the equation relating the Bell numbers to the Stirling numbers, each partition counted on the left hand side of the equation is counted in exactly one of the terms of the sum on the right hand side, the one for which k is the number of sets in the partition. [8] Spivey 2008 has given a formula that combines both of these summations:
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 into parts not divisible by is equal to the number of partitions in which no part is repeated or more times.