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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]
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
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.
Sigma function: Sums of powers of divisors of a given natural number. Euler's totient function: Number of numbers coprime to (and not bigger than) a given one. Prime-counting function: Number of primes less than or equal to a given number. Partition function: Order-independent count of ways to write a given positive integer as a sum of positive ...
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
The numbers within the triangle count partitions in which a given element is the largest singleton. The number of partitions of an n-element set into exactly k (non-empty) parts is the Stirling number of the second kind S(n, k). The number of noncrossing partitions of an n-element set is the Catalan number