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Séminaire Lotharingien de Combinatoire, B54b (2005), 10 pp.

# Ira M. Gessel

# Symmetric Inclusion-Exclusion

**Abstract.**
One form of the inclusion-exclusion principle asserts that if *A* and *B* are functions of finite sets then the formulas
and
are equivalent.
If we replace *B*(*S*)$ by
(-1)^{|S|}*B*(*S*)
then these formulas take on the symmetric form

which we call *symmetric inclusion-exclusion*. We study instances of symmetric inclusion-exclusion in which the functions *A* and *B* have combinatorial or probabilistic interpretations. In particular, we study cases related to the Pólya-Eggenberger urn model in which *A*(*S*) and *B*(*S*) depend only on the cardinality of *S*.

Received: May 12, 2005.
Accepted: July 20, 2005.
Final Version: July 23, 2005.

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