partition of a set

A probability distribution can be viewed as a partition of a set.

These polynomials have a remarkable combinatorics interpretation: the coefficients count certain partition of a set.

Likewise, intersections of elements from a sigma algebra result in a partition of a set.

The two definitions are equivalent: The unions of elements from a partition of a set generate a sigma algebra.

Disjoint-set data structures model the partitioning of a set, for example to keep track of the connected components of an undirected graph.

The AMI is defined in analogy to the adjusted Rand index of two different partitions of a set.

If a set is Partition of a set into Equivalence class, then one member can be chosen from each equivalence class to represent that class.

Today, Dobinski's formula is sometimes stated by saying the number of partitions of a set of size n equals the nth moment of that distribution.

In other words, the nth moment of this probability distribution is the number of partitions of a set of size n into no more than m parts.

In mathematics, space partitioning is the process of dividing a space (usually a Euclidean space) into two or more disjoint subsets (see also partition of a set).

The Stirling numbers of the second kind, S(n,k) count the number of partitions of a set of n elements into k non-empty subsets (indistinguishable boxes).

This guarantees for any partition of a set X the existence of a subset C of X containing exactly one element from each part of the partition.

The value at 1 of the nth Touchard polynomial is the nth Bell number, i.e., the number of partitions of a set of size n:

Whereas the modern atom is indeed divisible, it actually is uncuttable: there is no Partition of a set of space such that its parts correspond to material parts of the atom.

In combinatorics, the nth Bell number, named after Eric Temple Bell, is the number of partitions of a set with n members, or equivalently, the number of equivalence relations on it.

A partition of a set X is a set of nonempty subsets of X such that every element x in X is in exactly one of these subsets.

The "coefficient" in each term is the number of partitions of a set of "n" members that collapse to that partition of the integer "n" when the members of the set become indistinguishable.

The coefficient 6 that goes with those factors corresponds to the fact that there are exactly six partitions of a set of four members that break it into one part of size 2 and two parts of size 1.

Just as order relations are grounded in ordered sets, sets closed under pairwise supremum and infimum, equivalence relations are grounded in partition of a set, sets closed under bijections preserving partition structure.

In mathematics, in the area of order theory, 'Dilworth's theorem' characterizes the 'width' of any finite partially ordered set in terms of a Partition of a set of the order into a minimum number of chains.

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 particular, the nth moment of the Poisson distribution with expected value 1 is precisely the number of partitions of a set of size n, i.e., it is the nth Bell number (this fact is Dobinski's formula).

Unlike the lattice of all partitions of the set, the lattice of all noncrossing partitions of a set is self-dual, i.e., it is order-isomorphic to the lattice that results from inverting the partial order ("turning it upside-down").

The Bell numbers, named after Eric Temple Bell, count the number of partitions of a set, and the weak orderings that are counted by the ordered Bell numbers may be interpreted as a partition together with a total order on the sets in the partition.

Consequently; 42 is the number of noncrossing partitions of a set of five elements, the number of triangulations of a heptagon, the number of rooted ordered binary trees with six leaves, the number of ways in which five pairs of nested parentheses can be arranged, etc.