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A combinatorial bound involving Stirling numbers of the second type

My previous question was solved in a very elegant way, hopefully this (seemingly more complicated) case is also easy for experts. I need the inequality ...
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Stirling numbers of first kind over multiset

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 $$ ...