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} Combinatorics } Stirling Numbers of the 4th Kind?

18/11/2013 - 21:00 von TheInquirer | Report spam
Define <n,k> := number of ways to throw n distinct (numbered) balls
into k distinct (numbered) non-empty bins [there must be at least one
ball in each bin. the order of the balls within each bin is immaterial.]

It is easy to see that
<n,0> = Kronecker_delta(n,0),
and we also have the recurrence relation
<n+1,k> = k(<n,k> + <n,k-1>)


what are these numbers <n,k>?
Stirling Numbers of the 4th Kind?
How are they related to the other kinds of Stirling Numbers?
any literature on this kind of numbers?
 

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#1 TheInquirer
19/11/2013 - 20:50 | Warnen spam
On 11/19/2013 4:00 AM, TheInquirer wrote:
Define <n,k> := number of ways to throw n distinct (numbered) balls
into k distinct (numbered) non-empty bins [there must be at least one
ball in each bin. the order of the balls within each bin is immaterial.]

It is easy to see that
<n,0> = Kronecker_delta(n,0),
and we also have the recurrence relation
<n+1,k> = k(<n,k> + <n,k-1>)


what are these numbers <n,k>?
Stirling Numbers of the 4th Kind?
How are they related to the other kinds of Stirling Numbers?
any literature on this kind of numbers?




I just found out that <n,k> = {n,k}*k!
where {n,k} is Stirlings number of the second kind.

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