Inclusion-exclusion, cancellation, and consecutive sets

The nnnrlnip of inclusion and exclusion and Consecutive Sets M. H. McIlwain )p'n,Clrj·'Yli'·nl of Mathematics Furman University Greenville, SC USA 29613 applied to numerous areas of discrete and combinatorial mathematics. One manifestation of this occurs in probability of the union of events an alternating sum of pn)o(HHjlIW~S of various intersections of events. If the constituent events are themselves Sult1lC:lerltlv well structured, then predictable cancellation occurs in this expansion. We discuss the special case in which each of the underlying sets is "consecutive": namely, its elements are consecutive integers. For such consecutive systems the inclusionexclusion expansion assumes a particularly simple form, in which all reduced codficients in the expression equal ± 1. Moreover, the appropriate sign of each noncancelling term is dictated by the length of a certain path in a graph derived from the incidence structure of the given sets.