Abstract
Let (A_v)_v ∈ V be a finite family of sets. We establish an improved inclusion-exclusion identity for each closure operator on the power set of V having the unique base property. The result generalizes three improvements of the inclusion-exclusion principle as well as Whitney's broken circuit theorem on the chromatic polynomial of a graph.
Highlights
One of the most important tools in enumerative combinatorics and combinatorial probability theory is the principle of inclusion-exclusion, which is known as the sieve formula or the formula of Poincareor Sylvester
In the proof of his famous inclusion-exclusion variant for semilattices, Narushima [15] uses a prominent result of Rota [16] on closure operators and Mobius functions
✂ ✄✆✝✂✤✞ ✁ Theorem 1 Let ☞✍✌ ✎ ✑ ✎✕✔✗✖ be a finite family of sets and be a closure operator on ☞
Summary
One of the most important tools in enumerative combinatorics and combinatorial probability theory is the principle of inclusion-exclusion, which is known as the sieve formula or the formula of Poincareor Sylvester. In the proof of his famous inclusion-exclusion variant for semilattices, Narushima [15] uses a prominent result of Rota [16] on closure operators and Mobius functions. A closure operator is said to have the unique base property if none of its closed sets has more than one base.
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