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The rank of a partition is the largest number k such that the partition contains at least k parts of size at least k. For example, the partition 4 + 3 + 3 + 2 + 1 + 1 has rank 3 because it contains 3 parts that are ≥ 3, but does not contain 4 parts that are ≥ 4.
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]
In combinatorial mathematics, Dobiński's formula [1] states that the th Bell number, the number of partitions of a set of size , equals = =!, where denotes Euler's number. The formula is named after G. Dobiński, who published it in 1877.
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
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. [17] It is denoted by S r ( n , k ) {\displaystyle S_{r}(n,k)} and obeys the recurrence relation
Partitions of sets can be arranged in a partial order, showing that each partition of a set of size n "uses" one of the partitions of a set of size n − 1. The 52 partitions of a set with 5 elements. In general, is the number of partitions of a set of size . A partition of a set is defined as a family of nonempty, pairwise disjoint subsets of ...
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 52 5 = 10 + 2 5 {\textstyle {\frac {52}{5}}=10+{\frac {2}{5}}} .
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]