Stirling number
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Stirling numbers of the first kind
In combinatorics, unsigned Stirling numbers of the first kind
- <math>s(n,k)\,<math>
(with a lower-case "s") count the number of permutations of n elements with k disjoint cycles.
Stirling numbers of the first kind (without the qualifying adjective unsigned) are the coefficients in the expansion
- <math>x^n=\sum_{k=1}^n s(n,k)(x)^k<math>
where (x)n is the rising factorial
- <math>(x)^n=x(x+1)(x+2)\cdots(x+n-1).<math>
Stirling numbers of the first kind are sometimes written with the alternate notation
- <math>s(n,k)=\left[\begin{matrix} n \\ k \end{matrix}\right].<math>
The definition can be inverted to express the falling factorial as a power series:
- <math>(x)_n = \sum_{k=0}^n s(n,k) x^k<math>
Many relations for the Stirling numbers shadow similar relations on the binomial coefficients. The study of these 'shadow relationships' is termed umbral calculus and culminates in the theory of Sheffer sequences.
Stirling numbers of the first kind can be expressed in terms of the harmonic numbers as follows:
- <math>s(n,k)=(-)^{k-n} \frac{\Gamma(n)}{\Gamma(k)}w(n,k-1)<math>
where <math>w(n,0)=1<math> and
- <math>w(n,k)=\sum_{m=0}^{k-1}\frac{\Gamma(1-k+m)}{\Gamma(1-k)}H_{n-1}^{(m+1)} w(n,k-1-m)<math>
In the above, <math>\Gamma(x)<math> is the Gamma function and <math>H^{(m)}_n<math> is the harmonic number.
Stirling numbers of the second kind
Stirling numbers of the second kind S(n,k) (with a capital "S") count the number of equivalence relations having k equivalence classes defined on a set with n elements. The sum
- <math>B_n=\sum_{k=1}^n S(n,k)<math>
is the nth Bell number. If we let
- <math>(x)_n=x(x-1)(x-2)\cdots(x-n+1)<math>
(in particular, (x)0 = 1 because it is an empty product) be the falling factorial, we can characterize the Stirling numbers of the second kind by
- <math>\sum_{k=1}^n S(n,k)(x)_k=x^n.<math>
(Confusingly, the notation that combinatorialists use for falling factorials coincides with the notation used in special functions for rising factorials; see Pochhammer symbol.) The Stirling numbers of the second kind enjoy the following relationship with the Poisson distribution: if X is a random variable with a Poisson distribution with expected value λ, then its nth moment is
- <math>E(X^n)=\sum_{k=1}^n S(n,k)\lambda^k.<math>
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").
Karamata notation
In recent years, the Stirling numbers have often been denoted in a way introduced in 1935 by Jovan Karamata and promoted later by Donald Knuth:
- <math>s(n,k)=\left[\begin{matrix} n \\ k \end{matrix}\right].<math>
- <math>S(n,k)=\left\{\begin{matrix} n \\ k \end{matrix}\right\}.<math>
See also
External references
- D.E. Knuth, Two notes on notation (http://www-cs-faculty.stanford.edu/~knuth/papers/tnn.tex.gz) (TeX source).pl:Liczby Stirlinga