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  2. Cartesian product - Wikipedia

    en.wikipedia.org/wiki/Cartesian_product

    A table can be created by taking the Cartesian product of a set of rows and a set of columns. If the Cartesian product rows × columns is taken, the cells of the table contain ordered pairs of the form (row value, column value). [4] One can similarly define the Cartesian product of n sets, also known as an n-fold Cartesian product, which can be ...

  3. Partition of a set - Wikipedia

    en.wikipedia.org/wiki/Partition_of_a_set

    Partitions of a 4-element set ordered by refinement. A partition α of a set X is a refinement of a partition ρ of X—and we say that α is finer than ρ and that ρ is coarser than α—if every element of α is a subset of some element of ρ. Informally, this means that α is a further fragmentation of ρ. In that case, it is written that ...

  4. Stirling numbers of the second kind - Wikipedia

    en.wikipedia.org/wiki/Stirling_numbers_of_the...

    An r-associated Stirling number of the second kind is the number of ways to partition a set of n objects into k subsets, with each subset containing at least r elements. [17] It is denoted by S r ( n , k ) {\displaystyle S_{r}(n,k)} and obeys the recurrence relation

  5. List of partition topics - Wikipedia

    en.wikipedia.org/wiki/List_of_partition_topics

    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

  6. Bell number - Wikipedia

    en.wikipedia.org/wiki/Bell_number

    Partitions of sets can be arranged in a partial order, showing that each partition of a set of size n "uses" one of the partitions of a set of size n − 1. The 52 partitions of a set with 5 elements. In general, is the number of partitions of a set of size . A partition of a set is defined as a family of nonempty, pairwise disjoint subsets of ...

  7. Pentagonal number theorem - Wikipedia

    en.wikipedia.org/wiki/Pentagonal_number_theorem

    The theorem can be interpreted combinatorially in terms of partitions. In particular, the left hand side is a generating function for the number of partitions of n into an even number of distinct parts minus the number of partitions of n into an odd number of distinct parts.

  8. Multiway number partitioning - Wikipedia

    en.wikipedia.org/wiki/Multiway_number_partitioning

    There is an optimal partition in which all subsets sums are strictly betweel L/2 and 2L (L/2 < C i < 2L for all i in 1,...,k). Particularly, the partition minimizing the sum of squares C i 2, among all optimal partitions, satisfies these inequalities. The PTAS uses an input rounding technique.

  9. Partition problem - Wikipedia

    en.wikipedia.org/wiki/Partition_problem

    An instance of SubsetSum consists of a set S of positive integers and a target sum T; the goal is to decide if there is a subset of S with sum exactly T. Given such an instance, construct an instance of Partition in which the input set contains the original set plus two elements: z 1 and z 2, with z 1 = sum(S) and z 2 = 2T.