Generally, a partition is a division of a whole into non-overlapping parts. Among the kinds of partitions considered in

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

are *

partition of a set In mathematics, a partition of a set is a grouping of its elements into Empty set, non-empty subsets, in such a way that every element is included in exactly one subset. Every equivalence relation on a Set (mathematics), set defines a partitio ...

or an ordered partition of a set, * partition of a graph, * partition of an integer, *

partition of an interval In mathematics, a partition of an interval on the real line is a finite sequence of real numbers such that :. In other terms, a partition of a compact interval is a strictly increasing sequence of numbers (belonging to the interval itsel ...

, *

partition of unity In mathematics, a partition of unity on a topological space is a Set (mathematics), set of continuous function (topology), continuous functions from to the unit interval ,1such that for every point x\in X: * there is a neighbourhood (mathem ...

, * partition of a matrix; see

block matrix In mathematics, a block matrix or a partitioned matrix is a matrix that is interpreted as having been broken into sections called blocks or submatrices. Intuitively, a matrix interpreted as a block matrix can be visualized as the original matrix w ...

, and * partition of the sum of squares in

statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...

problems, especially in the

analysis of variance Analysis of variance (ANOVA) is a family of statistical methods used to compare the Mean, means of two or more groups by analyzing variance. Specifically, ANOVA compares the amount of variation ''between'' the group means to the amount of variati ...

, *

quotition and partition In arithmetic, quotition and partition are two ways of viewing fractions and division. In quotitive division one asks "how many parts are there?" while in partitive division one asks "what is the size of each part?" In general, a quotient Q = N / ...

, two ways of viewing the operation of division of integers.

Integer partitions

Composition (combinatorics) In mathematics, a composition of an integer ''n'' is a way of writing ''n'' as the summation, sum of a sequence of (strictly) positive integers. Two sequences that differ in the order of their terms define different compositions of their sum, whi ...

* Ewens's sampling formula * Ferrers graph * Glaisher's theorem * Landau's function *

Partition function (number theory) In number theory, the partition function represents the number of possible partitions of a non-negative integer . For instance, because the integer 4 has the five partitions , , , , and . No closed-form expression for the partition function i ...

Pentagonal number theorem In mathematics, Euler's pentagonal number theorem relates the product and series representations of the Euler function. It states that :\prod_^\left(1-x^\right)=\sum_^\left(-1\right)^x^=1+\sum_^\infty(-1)^k\left(x^+x^\right). In other words, : ...

Plane partition In mathematics and especially in combinatorics, a plane partition is a two-dimensional array of nonnegative integers \pi_ (with positive number, positive integer indices ''i'' and ''j'') that is nonincreasing in both indices. This means that : \pi ...

Quotition and partition In arithmetic, quotition and partition are two ways of viewing fractions and division. In quotitive division one asks "how many parts are there?" while in partitive division one asks "what is the size of each part?" In general, a quotient Q = N / ...

Rank of a partition In number theory and combinatorics, the rank of an integer partition is a certain number associated with the partition. In fact at least two different definitions of rank appear in the literature. The first definition, with which most of this ar ...

Crank of a partition In number theory, the crank of an integer partition is a certain number associated with the partition. It was first introduced without a definition by Freeman Dyson, who hypothesised its existence in a 1944 paper. Dyson gave a list of properties ...

Solid partition Solid is a state of matter where molecules are closely packed and can not slide past each other. Solids resist compression, expansion, or external forces that would alter its shape, with the degree to which they are resisted dependent upon the ...

Young tableau In mathematics, a Young tableau (; plural: tableaux) is a combinatorial object useful in representation theory and Schubert calculus. It provides a convenient way to describe the group representations of the symmetric and general linear groups an ...

Young's lattice In mathematics, Young's lattice is a lattice that is formed by all integer partitions. It is named after Alfred Young, who, in a series of papers ''On quantitative substitutional analysis,'' developed the representation theory of the symmetric ...

Set partitions

{{main, Partition of a set *

Bell number In combinatorial mathematics, the Bell numbers count the possible partitions of a set. These numbers have been studied by mathematicians since the 19th century, and their roots go back to medieval Japan. In an example of Stigler's law of epony ...

* Bell polynomials ** Dobinski's formula *

Cumulant In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will have ...

Data clustering Cluster analysis or clustering is the data analyzing technique in which task of grouping a set of objects in such a way that objects in the same group (called a cluster) are more similar (in some specific sense defined by the analyst) to each o ...

Equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric, and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equ ...

Exact cover In the mathematical field of combinatorics, given a collection \mathcal of subsets of a set X, an exact cover is a subcollection \mathcal^ of \mathcal such that each element in X is contained in ''exactly one'' subset in \mathcal^. One says that e ...

** Knuth's Algorithm X *** Dancing Links * Exponential formula *

Faà di Bruno's formula Faà di Bruno's formula is an identity in mathematics generalizing the chain rule to higher derivatives. It is named after , although he was not the first to state or prove the formula. In 1800, more than 50 years before Faà di Bruno, the French ...

* Feshbach–Fano partitioning *

Foliation In mathematics (differential geometry), a foliation is an equivalence relation on an topological manifold, ''n''-manifold, the equivalence classes being connected, injective function, injectively immersed submanifolds, all of the same dimension ...

* Frequency partition *

Graph partition In mathematics, a graph partition is the reduction of a Graph (discrete mathematics), graph to a smaller graph by partition of a set, partitioning its set of nodes into mutually exclusive groups. Edges of the original graph that cross between the g ...

Kernel of a function In set theory, the kernel of a Function (mathematics), function f (or equivalence kernel.) may be taken to be either * the equivalence relation on the function's Domain of a function, domain that roughly expresses the idea of "equivalent as fa ...

* Lamination (topology) * Matroid partitioning * Multipartition *

Multiplicative partition In number theory, a multiplicative partition or unordered factorization of an integer n is a way of writing n as a product of integers greater than 1, treating two products as equivalent if they differ only in the ordering of the factors. The number ...

* Noncrossing partition * Ordered partition of a set * Partition calculus *

Partition function (quantum field theory) In quantum field theory, partition functions are generating functionals for correlation functions, making them key objects of study in the path integral formalism. They are the imaginary time versions of statistical mechanics partition functi ...

Partition function (statistical mechanics) In physics, a partition function describes the statistics, statistical properties of a system in thermodynamic equilibrium. Partition functions are function (mathematics), functions of the thermodynamic state function, state variables, such a ...

** Derivation of the partition function *

Partition of an interval In mathematics, a partition of an interval on the real line is a finite sequence of real numbers such that :. In other terms, a partition of a compact interval is a strictly increasing sequence of numbers (belonging to the interval itsel ...

Partition of a set In mathematics, a partition of a set is a grouping of its elements into Empty set, non-empty subsets, in such a way that every element is included in exactly one subset. Every equivalence relation on a Set (mathematics), set defines a partitio ...

** Ordered partition **

Partition refinement In the design of algorithms, partition refinement is a technique for representing a partition of a set as a data structure that allows the partition to be refined by splitting its sets into a larger number of smaller sets. In that sense it is dual t ...

Disjoint-set data structure In computer science, a disjoint-set data structure, also called a union–find data structure or merge–find set, is a data structure that stores a collection of Disjoint sets, disjoint (non-overlapping) Set (mathematics), sets. Equivalently, it ...

Partition problem In number theory and computer science, the partition problem, or number partitioning, is the task of deciding whether a given multiset ''S'' of positive integers can be partition of a set, partitioned into two subsets ''S''1 and ''S''2 such that th ...

** 3-partition problem *

Partition topology In mathematics, a partition topology is a topology that can be induced on any set X by partitioning X into disjoint subsets P; these subsets form the basis for the topology. There are two important examples which have their own names: * The is t ...

Recursive partitioning Recursive partitioning is a statistics, statistical method for multivariable analysis. Recursive partitioning creates a Decision tree learning, decision tree that strives to correctly classify members of the population by splitting it into sub-popu ...

Stirling number In mathematics, Stirling numbers arise in a variety of Analysis (mathematics), analytic and combinatorics, combinatorial problems. They are named after James Stirling (mathematician), James Stirling, who introduced them in a purely algebraic setti ...

Stirling transform In combinatorial mathematics, the Stirling transform of a sequence of numbers is the sequence given by :b_n=\sum_^n \left\ a_k, where \left\ is the Stirling number of the second kind, which is the number of partitions of a set of size n into k ...

Stratification (mathematics) Stratification has several usages in mathematics. In mathematical logic In mathematical logic, stratification is any consistent assignment of numbers to predicate symbols guaranteeing that a unique formal interpretation of a logical theory exis ...

* Tverberg partition * Twelvefold way

In probability and stochastic processes

* Chinese restaurant process * Dobinski's formula * Ewens's sampling formula * Law of total cumulance Partition Partition topics