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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.
In plain words, e.g., the first congruence means that If a number is 4 more than a multiple of 5, i.e. it is in the sequence 4, 9, 14, 19, 24, 29, . . . then the number of its partitions is a multiple of 5. Later other congruences of this type were discovered, for numbers and for Tau-functions.
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]
These two types of partition are in bijection with each other, by a diagonal reflection of their Young diagrams. Their numbers can be arranged into a triangle, the triangle of partition numbers , in which the n {\displaystyle n} th row gives the partition numbers p 1 ( n ) , p 2 ( n ) , … , p n ( n ) {\displaystyle p_{1}(n),p_{2}(n),\dots ,p ...
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
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: