Signed partitions - A 'balls into urns' approach

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 generaliza¬tion of a well-known identity concerning the classical Stirling numbers S(n, k) of the second kind: n xn = £ 5(n, k)-x{x-I)... (a? - A + 1). fc=o 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.