This paper considers the problem of enumeration under group actions in the framework of multilinear algebra. Vector identities are derived in a certain algebra of tensors. Various classes of combinatorial identities may be derived from these vector identities by evaluation with appropriately chosen linear functionals. Including among these combinatorial identities is Polya's enumeration theorem. The principal intent of this paper is to develop identities analogous to Polya's identity but which allow one to incorporate restrictions on the class of objects being enumerated in addition to those which can be treated by Polya's Theorem or its various existing extensions. In this paper, identities are developed to treat certain classes of ''local conditions'' in the structure being enumerated.
['S. G. Williamson']
The combinatorial analysis of patterns and the principle of inclusion-exclusion