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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 ...
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]
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 ...
1) The number of partitions whose parts differ by at least 2 is equal to the number of partitions involving only numbers congruent to 1 or 4 (mod 5). 2) The number of partitions whose parts differ by at least 2 and with the smallest part at least 2 is equal to the number of partitions involving only numbers congruent to 2 or 3 (mod 5).
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
For any pair of positive integers n and k, the number of k-tuples of positive integers whose sum is n is equal to the number of (k − 1)-element subsets of a set with n − 1 elements. For example, if n = 10 and k = 4, the theorem gives the number of solutions to x 1 + x 2 + x 3 + x 4 = 10 (with x 1, x 2, x 3, x 4 > 0) as the binomial coefficient
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
The following are not partitions of {1, 2, 3}: { {}, {1, 3}, {2} } is not a partition (of any set) because one of its elements is the empty set. { {1, 2}, {2, 3} } is not a partition (of any set) because the element 2 is contained in more than one block. { {1}, {2} } is not a partition of {1, 2, 3} because none of its blocks contains 3; however ...
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