Eli Bagno, David Garber: Signed partitions - A 'balls into urns' approach, 63-71

Abstract:

Using Reiner's definition of Stirling numbers of the second kind for the group of signed permutations, we provide a 'balls into urns' approach for proving a generalization of a well-known identity concerning the classical Stirling numbers $S(n,k)$ of the second kind:

$$x^n=\sum\limits_{k=0}^n{S(n,k)\cdot x(x-1)\cdots (x-k+1)}.$$

We also present a combinatorial proof (based on Feller's coupling) of the defining identity for the Stirling numbers of the first kind in the group of signed permutations.

Our proofs are self-contained and accessible also for non-experts.

Key Words: Stirling number, signed partitions, 'balls into urns' approach.

2010 Mathematics Subject Classification: Primary 05A18; Secondary 05A19.

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