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Given a multiset $$ M= \{ 1^{a1},2^{a2},…,k^{ak} \}\ where\ N=\Sigma_j\ aj $$

$ f(M,r)$ denotes the number of permutations of the multiset $M$ that have exactly $r$ strongly outstanding elements

$$ f(M,r)=(\ (\ ^N_{a1}\ )−(\ ^{N−1}_{a1−1}\ )\ )f(M/1^{a1},r)+(^{N−1}_{a1−1})f(M/1^{a1},r−1) $$

Is there a closed form expression for this 'Stirling numbers of first kind over multiset' recursion?

I know that there is a generator for

$$ F_M(x)=\Sigma _rf(M,r)x^r $$ which is the sum over all possible $r$ (but I want closed form expression for only a particular $r$).

[Encountered this when reading: http://www.math.upenn.edu/~wilf/website/LRMaxima.pdf ]

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A similar question is active in a codechef competition, see, e.g., math.stackexchange.com/questions/670815/… –  Gerry Myerson Feb 11 at 21:39

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