Search results
Results from the WOW.Com Content Network
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.
The initial idea is usually attributed to the work of Hardy with Srinivasa Ramanujan a few years earlier, in 1916 and 1917, on the asymptotics of the partition function.It was taken up by many other researchers, including Harold Davenport and I. M. Vinogradov, who modified the formulation slightly (moving from complex analysis to exponential sums), without changing the broad lines.
Among the 22 partitions of the number 8, there are 6 that contain only odd parts: 7 + 1; 5 + 3; 5 + 1 + 1 + 1; 3 + 3 + 1 + 1; 3 + 1 + 1 + 1 + 1 + 1; 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1; Alternatively, we could count partitions in which no number occurs more than once. Such a partition is called a partition with distinct parts. If we count the ...
It is seen to have dimension 0 only in the cases where ℓ = 5, 7 or 11 and since the partition function can be written as a linear combination of these functions [4] this can be considered a formalization and proof of Ramanujan's observation.
The partition function or configuration integral, as used in probability theory, information theory and dynamical systems, is a generalization of the definition of a partition function in statistical mechanics. It is a special case of a normalizing constant in probability theory, for the Boltzmann distribution.
In mathematics, the Hardy–Ramanujan theorem, proved by Ramanujan and checked by Hardy [1] states that the normal order of the number () of distinct prime factors of a number is . Roughly speaking, this means that most numbers have about this number of distinct prime factors.
Let n be a non-negative integer and let p(n) denote the number of partitions of n (p(0) is defined to be 1).Srinivasa Ramanujan in a paper [3] published in 1918 stated and proved the following congruences for the partition function p(n), since known as Ramanujan congruences.
then the partition λ is called a strict partition of n. The integers λ k, λ k − 1, ..., λ 1 are the parts of the partition. The number of parts in the partition λ is k and the largest part in the partition is λ k. The rank of the partition λ (whether ordinary or strict) is defined as λ k − k. [1]