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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 ...
Garvan's thesis, Generalizations of Dyson's rank, concerned the rank of a partition [3] and formed the groundwork for several of his later papers. [4] Garvan is well-known for his work in the fields of q-series and integer partitions. Most famously, in 1988, Garvan and Andrews discovered a definition of the crank of a partition. [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.
In 1944, Freeman Dyson defined the rank function for a partition and conjectured the existence of a "crank" function for partitions that would provide a combinatorial proof of Ramanujan's congruences modulo 11.
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,
The values (), …, of the partition function (1, 2, 3, 5, 7, 11, 15, and 22) can be determined by counting the Young diagrams for the partitions of the numbers from 1 to 8. In number theory, the partition function p(n) represents the number of possible partitions of a non-negative integer n.