In this paper, we consider a two-dimensional model for finite set partitions which arises in conjunction with a special case of a general non-linear recurrence. We investigate properties of some of the related counting sequences, including recurrences and generating functions. In particular, we obtain, by combinatorial arguments, some formulas relating these sequences to the Stirling numbers of the first kind. Specializing these arguments yields bijective proofs of some recent identities of Gould and Quaintance involving the Bell numbers, which were established using algebraic methods.