Parity structures and generating functions from Boolean rings

David Petrie Moulton¹
Department of Mathematics
University of Wisconsin
Madison, WI 53706
- William A. Stein
Department of Mathematics
University of California
Berkeley, CA 94720

October 27, 1999

Abstract:

Let S be a finite set and T be a subset of the power set of S. Call T a parity structure for S if, for each subset b of S of odd size, the number of subsets of b that lie in T is even. We classify parity structures using generating functions from a free boolean ring. We also show that if T is a parity structure, then, for each subset b of S of even size, the number of subsets of b of odd size that lie in T is even. We then give several other properties of parity structures and discuss a generalization.