ar X iv : 0 80 8 . 23 07 v 1 [ m at h . C O ] 1 7 A ug 2 00 8 BOOLEAN COMPLEXES FOR FERRERS GRAPHS

Ferrers graphs are bipartite graphs that correspond naturally to Ferrers shapes. In this paper, we determine the homotopy type of the boolean complex of Ferrers graphs. Previous work of the last two authors shows that the boolean complex of a graph is a wedge of spheres of maximal dimension. Thus the homotopy type of this complex depends entirely on the number of spheres in the wedge sum, called the boolean number of the Ferrers graph. Applying the results 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.