Arrangements on chessboards

For the study of the distribution of like objects on chessboards, i.e., rectangular arrays of square cells, with no lines (rows or columns) of the boards empty, the enumerators Ar, s(t) = ∑tmA(m; r, s) of r by s rectangles by number of objects, defined by (1+t)xy∑r,s=0(xr)(ys)Ar,s(t) are shown to hold a central position. Their properties and uses are examined at length.