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The order-dependent composition 1 + 3 is the same partition as 3 + 1, and the two distinct compositions 1 + 2 + 1 and 1 + 1 + 2 represent the same partition as 2 + 1 + 1. An individual summand in a partition is called a part. The number of partitions of n is given by the partition function p(n). So p(4) = 5.
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.
Analogously to Pascal's triangle, these numbers may be calculated using the recurrence relation [2] = + (). As base cases, p 1 ( n ) = 1 {\displaystyle p_{1}(n)=1} , and any value on the right hand side of the recurrence that would be outside the triangle can be taken as zero.
The sums of each row of the triangle, 1, 3, 10, 37, ..., are the same sequence of first differences appearing in the second-from-right diagonal of the triangle. [5] The nth number in this sequence also counts the number of partitions of n elements into subsets, where one of the subsets is distinguished from the others; for instance, there are ...
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 the number of sets in the partition. [8] Spivey 2008 has given a formula that combines both of these summations:
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. Although the partition problem is NP-complete, there is a ...
If there is a remainder in solving a partition problem, the parts will end up with unequal sizes. For example, if 52 cards are dealt out to 5 players, then 3 of the players will receive 10 cards each, and 2 of the players will receive 11 cards each, since = +.
The partition problem - a special case of multiway number partitioning in which the number of subsets is 2. The 3-partition problem - a different and harder problem, in which the number of subsets is not considered a fixed parameter, but is determined by the input (the number of sets is the number of integers divided by 3).