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On partitions, surjections, and Stirling numbers. (English) Zbl 0812.05007
It is proved that if \(S(m,n)\) denotes the Stirling number of the second kind then \[ S(m, m- k)= \sum^{k-1}_{h= 0} a_{hk}\begin{pmatrix} m\\ k+ h+ 1\end{pmatrix}, \] where the \(a_{hk}\) are positive integers, independent of \(m\), given inductively by \[ a_{0k}= 1\quad\text{and}\quad a_{hk}= \sum^{k-1}_{ j= h} \begin{pmatrix} k+ h\\ j+ h\end{pmatrix} a_{h- 1,j}. \] Various identities involving binomial coefficients and the numbers \(a_{hk}\) are obtained. Using the recurrence \(S(m,n)= nS(m- 1, n)+ S(m- 1,n- 1)\), it is shown that \[ \sum^{k-1}_{h= 0} \begin{pmatrix} n+ k-1\\ k+h\end{pmatrix} a_{hk}= n \sum^{k-2}_{h= 0} a_{h,k -1} \begin{pmatrix} n+ k-1\\ k+ h\end{pmatrix}. \] {}.
05A19 Combinatorial identities, bijective combinatorics
05A18 Partitions of sets
11B73 Bell and Stirling numbers
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