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Author: William A. Stein
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Parity structures and generating functions from Boolean rings
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# Parity structures and generating functions from Boolean rings

David Petrie Moulton1
Department of Mathematics
University of Wisconsin
- 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.

William Arthur Stein
1999-10-27