  In this paper we provide an explicit formula for calculating the boolean
number of a Ferrers graph. By previous work of the last two authors, this
determines the homotopy type of the boolean complex of the graph. Specializing
to staircase shapes, we show that the boolean numbers of the associated Ferrers
graphs are the Genocchi numbers of the second kind, and obtain a relation
between the Legendre-Stirling numbers and the Genocchi numbers of the second
kind. In another application, we compute the boolean number of a complete
bipartite graph, corresponding to a rectangular Ferrers shape, which is
expressed in terms of the Stirling numbers of the second kind. Finally, we
analyze the complexity of calculating the boolean number of a Ferrers graph
using these results and show that it is a significant improvement over
calculating by edge recursion.
