On sums over partially ordered sets
The author establishes a general theorem for reducing sums of type \(\sum_{y\geq x}g(y)\), where \(g\) is a mapping from a partially ordered set into an abelian group. The result is applied to deduce the known result of G.-C. Rota, as well as of \textit{A. Blass} and \textit{B. E. Sagan} [Adv. Math. 127, No. 1, 94-123 (1997; Zbl 0872.06004)] on the Möbius function, to improve the inclusion-exclusion principle (see also the author [Arch. Math. 72, No. 4, 298-303 (1999; Zbl 0934.05011)]), and to obtain some conclusions about the Tutte polynomial and Crapo's beta invariant in matroid theory.
scientific article; zbMATH DE number 1345964