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The rank of a partition, shown as its Young diagram Freeman Dyson in 2005 In number theory and combinatorics , the rank of an integer partition is a certain number associated with the partition. In fact at least two different definitions of rank appear in the literature.
The rank of a partition is the largest number k such that the partition contains at least k parts of size at least k. For example, the partition 4 + 3 + 3 + 2 + 1 + 1 has rank 3 because it contains 3 parts that are ≥ 3, but does not contain 4 parts that are ≥ 4.
The rank of a partition is the integer obtained by subtracting the number of parts in the partition from the largest part in the partition. For example, the rank of the partition λ = { 4, 2, 1, 1, 1 } of 9 is 4 − 5 = −1. Denoting by N(m, q, n), the number of partitions of n whose ranks are congruent to m modulo q, Dyson considered N(m, 5 ...
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]
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. [18] It is denoted by S r ( n , k ) {\displaystyle S_{r}(n,k)} and obeys the recurrence relation
Partition function (number theory) Pentagonal number theorem; Plane partition; R. Rank of a partition; Representation theory of the symmetric group; Rogers ...
In number theory, an ordered multiplicative partition of a positive integer is a representation of the number as a product of one or more of its divisors. For instance, 30 has 13 multiplicative partitions, as a product of one divisor (30 itself), two divisors (for instance 6 · 5 ), or three divisors ( 3 · 5 · 2 , etc.).
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