Cycle-counting p, q-rook theory and the p, q, y-Stirling numbers of the second kind (preliminary report)
by
Frederick Butler
West Virginia University

In this preliminary report, we generalize several previously studied versions of rook theory by defining the cycle-counting p, q-rook numbers. In this model, we weight each rook placement by an expression involving the parameters p, q, and y, and several statistics on the placement. After the basic definitions and results are presented, we use the cycle-counting p, q-rook numbers of the triangular board to define the p, q, y-Stirling numbers of the second kind. We note how the basic cycle-counting p, q-rook theory results reduce in this case, giving versions of known theorems for the classical Stirling numbers of the second kind. Finally we discuss how the rook placement statistics simplify in this case; some have easy descriptions in terms of set partitions.

Date received: February 6, 2005