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The Jacobi-Stirling numbers. (English) Zbl 1261.11022

This paper discusses the Jacobi-Stirling numbers, which are natural analogues of the classical Stirling numbers. For the triangular recurrence relation: \(S(n,k)=S(n-1,k-1) + W(n,k)S(n-1,k)\), it is well known that the Stirling numbers of the second kind have \(W(n,k)=k\), while the Stirling numbers of the first kind have \(W(n,k)=n-1\). In this paper it is shown that the Jacobi-Stirling numbers have \(W(n,k)\) being a quadratic function: \[ \left \{ {n \atop j} \right \}_\gamma = \left \{ {n-1 \atop j-1} \right \}_\gamma + j(j+2\gamma-1) \left \{ {n-1 \atop j} \right \}_\gamma, \] and \[ \left [ {n \atop j} \right ]_\gamma = \left [ {n-1 \atop j-1} \right ]_\gamma + (n-1)(n+2\gamma-2) \left [ {n-1 \atop j} \right ]_\gamma, \] where \(\left \{ {n \atop j} \right \}_\gamma\), \(\left [ {n \atop j} \right ]_\gamma\) are the Jacobi-Stirling numbers of the second kind, the first kind, respectively.
Furthermore, the authors give several combinatorial interpretations of the Jacobi-Stirling numbers. For all \(n, j, \gamma \in \mathbb{N} \cup \{0\}\), \(\left \{ {n \atop j} \right \}_\gamma\) is the number of Jacobi-Stirling set partitions of \([n]_2\) into \(\gamma\) zero blocks and \(j\) nonzero blocks. \(\left [ {n \atop j} \right ]_\gamma\) is the number of balanced Jacobi-Stirling permutation pairs \((\pi_1, \pi_2)\) of length \(n\) in which \(\pi_1\) has exactly \(\gamma+j\) cycles, or the number of unbalanced Jacobi-Stirling permutation pairs \((\pi_1, \pi_2)\) of length \(n\) in which \(\pi_2\) has exactly \(j\) cycles.

MSC:

11B73 Bell and Stirling numbers
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