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In the case of the number 4, partitions 4 and 1 + 1 + 1 + 1 are conjugate pairs, and partitions 3 + 1 and 2 + 1 + 1 are conjugate of each other. Of particular interest are partitions, such as 2 + 2, which have themselves as conjugate. Such partitions are said to be self-conjugate. [7] Claim: The number of self-conjugate partitions is the same ...
partition of a graph, partition of an integer, partition of an interval, partition of unity, partition of a matrix; see block matrix, and; partition of the sum of squares in statistics problems, especially in the analysis of variance, quotition and partition, two ways of viewing the operation of division of integers.
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 ...
Because it is the fourth power of a prime, 81 has the same number (five) of multiplicative partitions as 4 does of additive partitions. The number 30 has five multiplicative partitions: 2 × 3 × 5 = 2 × 15 = 6 × 5 = 3 × 10 = 30.
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]
A partition of n has a Durfee square of size s if s is the largest number such that the partition contains at least s parts with values ≥ s. [1] 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 number of partitions of such that the adjacent parts differ by at least 2 is the same as the number of partitions of such that each part is congruent to either 1 or 4 modulo 5. The number of partitions of such that the adjacent parts differ by at least 2 and such that the smallest part is at least 2 is the same as the number of partitions ...
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