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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 ...
An equivalent, but more visual, definition is that the Durfee square is the largest square that is contained within a partition's Ferrers diagram. [2] The side-length of the Durfee square is known as the rank of the partition. [3] The Durfee symbol consists of the two partitions represented by the points to the right or below the Durfee square.
The poset Y is graded: the minimal element is ∅, the unique partition of zero, and the partitions of n have rank n. This means that given two partitions that are comparable in the lattice, their ranks are ordered in the same sense as the partitions, and there is at least one intermediate partition of each intermediate rank. The poset Y is a ...
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 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]
Pages in category "Integer partitions" ... Partition function (number theory) Pentagonal number theorem; Plane partition; R. Rank of a partition;