Touchard polynomial
التعريفات والمعاني
== English ==
=== Etymology ===
Studied by Jacques Touchard.
=== Noun ===
Touchard polynomial (plural Touchard polynomials)
(mathematics) Any member of a polynomial sequence of binomial type defined by
T
n
(
x
)
=
∑
k
=
0
n
S
(
n
,
k
)
x
k
=
∑
k
=
0
n
{
n
k
}
x
k
,
{\displaystyle T_{n}(x)=\sum _{k=0}^{n}S(n,k)x^{k}=\sum _{k=0}^{n}\left\{{n \atop k}\right\}x^{k},}
where
S
(
n
,
k
)
=
{
n
k
}
{\displaystyle S(n,k)=\left\{{n \atop k}\right\}}
is a Stirling number of the second kind, i.e. the number of partitions of a set of size n into k disjoint non-empty subsets.