The 15 partitions of a 4-element set ordered in a Hasse diagram There are S(4,1), ..., S(4, 4) = 1, 7, 6, 1 partitions containing 1, 2, 3, 4 sets.
In mathematics, particularly in combinatorics, a Stirling number of the second kind (or Stirling partition number) is the number of ways to partition a set of n objects into k non-empty subsets and is denoted by or .[1] Stirling numbers of the second kind occur in the field of mathematics called combinatorics and the study of partitions. They are named after James Stirling.
The Stirling numbers of the first and second kind can be understood as inverses of one another when viewed as triangular matrices. This article is devoted to specifics of Stirling numbers of the second kind. Identities linking the two kinds appear in the article on Stirling numbers.
^Ronald L. Graham, Donald E. Knuth, Oren Patashnik (1988) Concrete Mathematics, Addison–Wesley, Reading MA. ISBN 0-201-14236-8, p. 244.
and 24 Related for: Stirling numbers of the second kind information
in combinatorics, a Stirling number ofthesecondkind (or Stirling partition number) is the number of ways to partition a set of n objects into k non-empty...
combinatorics, Stirlingnumbersofthe first kind arise in the study of permutations. In particular, theStirlingnumbersofthe first kind count permutations...
In mathematics, Stirlingnumbers arise in a variety of analytic and combinatorial problems. They are named after James Stirling, who introduced them in...
} where S ( n , ℓ ) {\displaystyle S(n,\ell )} denotes theStirlingnumbersofthesecondkind, and x ( ℓ ) = ( x ) ( x + 1 ) ⋯ ( x + ℓ − 1 ) {\displaystyle...
(x)^{(n)}=(x)(x+1)\cdots (x+n-1)} denotes the rising factorial, and S ( r , k ) {\displaystyle S(r,k)} denotes Stirlingnumbersofthesecondkind. Longest alternating subsequence...
The use of exponential generating functions (EGFs) to study the properties ofStirlingnumbers is a classical exercise in combinatorial mathematics and...
_{n=1}^{\infty }{\frac {(-1)^{n}}{n}}{s \choose n}.} TheStirlingnumbersofthesecondkind are given by the finite sum { n k } = 1 k ! ∑ j = 0 k ( − 1 ) k...
are theStirlingnumbersofthesecondkind, and n k _ = n ( n − 1 ) ⋯ ( n − k + 1 ) {\displaystyle n^{\underline {k}}=n(n-1)\cdots (n-k+1)} is the k {\displaystyle...
Aitken (1933). Touchard polynomials Catalan number Stirling number Stirlingnumbersofthe first kind Gardner 1978. Halmos, Paul R. (1974). Naive set theory...
where d = 365 and S2 are Stirlingnumbersofthesecondkind. Consequently, the desired probability is 1 − p0. This variation ofthe birthday problem is interesting...
Introducing an explicit expression for theStirlingnumbersofthesecondkind into the finite sum for the polylogarithm of nonpositive integer order (see above)...
2^{(1/4+o(1))n^{2}}} by results of Kleitman and Rothschild. Note that S(n, k) refers to Stirlingnumbersofthesecondkind. A relation R is called intransitive...
mathematics, the Bell polynomials, named in honor of Eric Temple Bell, are used in the study of set partitions. They are related to Stirling and Bell numbers. They...
set of n labeled elements: Note that S(n, k) refers to Stirlingnumbersofthesecondkind. The number of strict partial orders is the same as that of partial...
the generalized sum G s ( n , r ) {\displaystyle G_{s}(n,r)} when s ∈ N {\displaystyle s\in \mathbb {N} } is expanded by theStirlingnumbersofthe second...
Global Information
