Some remarks one the sieve formula, the Tutte polynomial and Crapo's beta invariant (Q1587842)

In different versions of the sieve formula or other alternating sums with a great number of different terms usually several terms cancel each others. The following nice observation gives a way to deal with these cancellations: Let \(V\) be a finite set, \(f\) and \(g\) be maps from \(P(V)\) into an additive group, such that \(f(I)=\sum_{J \supseteq I} g(J)\) for all \(I\subseteq V.\) Suppose that \(S\) is a union-closed family of subsets of \(V\), such that \(f(X)=0\) for any \(X\in S.\) Then for any \(I\subseteq \bigcap S\) we have \[ f(I)=\sum _{\substack{ J \supseteq I\\ J \not\supseteq X (\forall X \in S)}} g(J). \] Similar results are given to thin the sieve formula, the Tutte polynomial of matroids and Crapo's beta invariant with cancelling terms.