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Such a partition is called a partition with distinct parts. If we count the partitions of 8 with distinct parts, we also obtain 6: 8; 7 + 1; 6 + 2; 5 + 3; 5 + 2 + 1; 4 + 3 + 1; This is a general property. For each positive number, the number of partitions with odd parts equals the number of partitions with distinct parts, denoted by q(n).
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]
The Stirling number {} is the number of ways to partition a set of cardinality n into exactly k nonempty subsets. Thus, in the equation relating the Bell numbers to the Stirling numbers, each partition counted on the left hand side of the equation is counted in exactly one of the terms of the sum on the right hand side, the one for which k is ...
In number theory and computer science, the partition problem, or number partitioning, [1] is the task of deciding whether a given multiset S of positive integers can be partitioned into two subsets S 1 and S 2 such that the sum of the numbers in S 1 equals the sum of the numbers in S 2.
Their numbers can be arranged into a triangle, the triangle of partition numbers, in which the th row gives the partition numbers (), (), …, (): [1] k n
The Bell numbers themselves, on the left and right sides of the triangle, count the number of ways of partitioning a finite set into subsets, or equivalently the number of equivalence relations on the set. Sun & Wu (2011) provide the following combinatorial interpretation of each value in the triangle.
However, the coefficient of x 12 is −1 because there are seven ways to partition 12 into an even number of distinct parts, but there are eight ways to partition 12 into an odd number of distinct parts, and 7 − 8 = −1. This interpretation leads to a proof of the identity by canceling pairs of matched terms (involution method). [1]
[1]: sec.5 The problem is parametrized by a positive integer k, and called k-way number partitioning. [2] The input to the problem is a multiset S of numbers (usually integers), whose sum is k*T . The associated decision problem is to decide whether S can be partitioned into k subsets such that the sum of each subset is exactly T .