Order shattering and Wilsonʹs theorem Original Research Article

The notion of order shattering was introduced in Anstee et al. (Graphs Combin. 18 (2002) 59–73). Here, we pursue further the algebraic interpretation that was established there. With this tool we give a new proof and a generalization for Wilsonʹs theorem on the diagonal form for the incidence matrices of t-subsets vs. k-subsets (European J. Combin. 11 (1990) 609–615). This allows a generalization of the corresponding rank formula modulo p, where p is an arbitrary prime.