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In number theory and combinatorics, a partition of a non-negative integer n, also called an integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition. (If order matters, the sum becomes a composition.)
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.
In the number theory of integer partitions, the numbers () denote both the number of partitions of into exactly parts (that is, sums of positive integers that add to ), and the number of partitions of into parts of maximum size exactly .
Download QR code; Print/export Download as PDF; Printable version; In other projects ... Integer partition; C. Composition (combinatorics) Crank of a partition; D.
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
In his Eureka paper Dyson proposed the concept of the rank of a partition. 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.
In all cases, the largest sum in the LDM partition is at most times the optimum, and there are instances in which it is at least () times the optimum. For two-way partitioning, when inputs are uniformly-distributed random variables, the expected difference between largest and smallest sum is n − Θ ( log n ) {\displaystyle n^{-\Theta ...
Equal-cardinality partition is a variant in which both parts should have an equal number of items, in addition to having an equal sum. This variant is NP-hard too. [5]: SP12 Proof. Given a standard Partition instance with some n numbers, construct an Equal-Cardinality-Partition instance by adding n zeros. Clearly, the new instance has an equal ...