co.combinatorics - Derangements with repetition

Hi all,

given (a1,...,an) formed by distinct letters, it's a well known problem to count the number of permutations with no fixed element.

I've been trying to solve a generalization of this problem, when we allow repetition of the letters.

I was able only to partially solve the problem when we have only repetition of a single letter.

If we have n letters and only one of them is repeated p times, then the number O(n,p) of permutations with no fixed element is given by the following recursive relation:

$O(n,0)=O(n,1)=mbox{Derangement}(n)$

$O(p+1,p)=dots=O(2p-1,p)=0, O(2p,p)=p!$

$O(n,p)={n-pchoose p} p!sum_{k=0}^p{p choose k}O(n-p-k,p-k)$

Does anybody know where this problem have been studied before? Does anybody know a general solution for this problem?

Thanks in advance.

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