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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.
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
Composite number. Highly composite number; Even and odd numbers. Parity; Divisor, aliquot part. Greatest common divisor; Least common multiple; Euclidean algorithm; Coprime; Euclid's lemma; Bézout's identity, Bézout's lemma; Extended Euclidean algorithm; Table of divisors; Prime number, prime power. Bonse's inequality; Prime factor. Table of ...
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]
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 = +.
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 1 ...
In a paper [2] published in 1988 George E. Andrews and F. G. Garvan defined the crank of a partition as follows: For a partition λ, let ℓ(λ) denote the largest part of λ, ω(λ) denote the number of 1's in λ, and μ(λ) denote the number of parts of λ larger than ω(λ). The crank c(λ) is given by
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 of such that each part is congruent to either 2 or 3 modulo 5. Alternatively,