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Given a multiset $M = \{ 1^{a_1} , 2^{a_2} ,\ldots , k^{a_k} \}$ where $N = \sum_j a_j$

$f(M, r)$ denotes the number of permutations of the multiset $M$ that have exactly $r$ strongly outstanding elements $$ f(M, r) = (\binom{N}{a_1} - \binom{N - 1}{a_1 - 1}) f(M/1^{a_1},r) + \binom{N - 1}{a_1 - 1} f(M/1^{a_1}, r-1) $$

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

I know that there is a closed form expression for $F_M(x) = \sum_rf(M,r)x^r$ which is the sum over all possible $r$ (but I want closed form expression for only a paritcular $r$).

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

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The formula given in the reference isn't exactly a "closed form"... and getting the coefficient of $x^r$ from it looks daunting.

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  • $\begingroup$ that formula given in the reference is generating function. But what i needed is for a particular 'r' or sum for values of r in a given range, say [a, b]. $\endgroup$ – user92 Feb 11 '14 at 17:16

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