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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 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 partition function is a function of the temperature T and the microstate energies E 1, E 2, E 3, etc. The microstate energies are determined by other thermodynamic variables, such as the number of particles and the volume, as well as microscopic quantities like the mass of the constituent particles.
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.
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,
The partition function or configuration integral, as used in probability theory, information science and dynamical systems, is an abstraction of the definition of a partition function in statistical mechanics. In statistical mechanics, the partition function, Z, encodes the statistical properties of a system in thermodynamic equilibrium.
Denote the n objects to partition by the integers 1, 2, ..., n. Define the reduced Stirling numbers of the second kind, denoted S d ( n , k ) {\displaystyle S^{d}(n,k)} , to be the number of ways to partition the integers 1, 2, ..., n into k nonempty subsets such that all elements in each subset have pairwise distance at least d .
Each partition of n into an even number of distinct parts contributes +1 to the coefficient of x n; each partition into an odd number of distinct parts contributes −1. (The article on unrestricted partition functions discusses this type of generating function.) For example, the coefficient of x 5 is +1 because there are two