When evaluating the inclusion-exclusion expansion N0 −N(1) −N(2) − · · ·+N(1, 2) +N(1, 3) + · · · many of the terms N(· · ·) may turn out to be zero, and hence should be discarded beforehand. Often this can be done.

We present a new and elementary proof of some recent improvements of the classical inclusion-exclusion bounds. The key idea is to use an injective mapping, similar to the bijective mapping in Garsia and Milne’s “bijective” proof of the classical inclusion-exclusion principle.

Improved inclusion-exclusion identities via closure operators Klaus Dohmen Department of Computer Science, Humboldt-University Berlin, Unter den Linden 6, D-10099 Berlin, Germany E-mail: [email protected] received March 24, 1999, revised September 6, 1999, accepted April 15, 2000. Let be a finite family of sets. We establish an improved inclusion-exclusion identity for each closure...

One form of the inclusion-exclusion principle asserts that if A and B are functions of finite sets then the formulas A(S) = ∑ T⊆S B(T ) and B(S) = ∑ T⊆S(−1) |S|−|T A(T ) are equivalent. If we replace B(S) by (−1)B(S) then these formulas take on the symmetric form A(S) = ∑ T⊆S (−1) B(T ) B(S) = ∑ T⊆S (−1) A(T ). which we call symmetric inclusion-exclusion. We study instances of symmetric inclusi...

The theory of Fermion oscillators has two essential ingredients: zero-point energy and Pauli exclusion principle. We devlop the theory of the statistical mechanics of generalized q-deformed Fermion oscillator algebra with inclusion principle (i.e., without the exclusion principle), which corresponds to ordinary fermions with Pauli exclusion principle in the classical limit q → 1. Some of the re...