Parametric Möbius inversion formulas
The theory of the general Möbius function for the (incidence algebra) of locally finite posets \(P\) and its consequences for a myriad of applications in various aspects of enumerative and algebraic combinatorics in addition to number theory is well established by now. In this note, the author provides an interesting variant by letting the incidence algebra \(I(P)\) act naturally on \(F(P\times X)\), the space of all complex-valued functions on \(P\times X\), where \(X\) is a parameter set of interest, according to the rules \[ D_\xi f(a,x)=\xi f(a,x)= \sum_{a\leq b\in P}\xi(a,b)f(b,x), \] \[ D_\xi g(b,x)= \sum_{a\leq b\in P}g(a,x)\xi(a, b)=g\xi(b,x), \] which in turn permit inversion formulas (under the right circumstances) \[ g(a,x)= \sum_{a\leq b\in P}\xi(a,b)f(b,x)\quad\text{implies}\quad f(a,x)= \sum_{a\leq b\in P}\xi^{-1}(a,b)g(b,x), \] which may then be localized on \(\xi=\xi^{-1}=\mu\) and which for special combinations such as \(X=P=\mathbb{N}\), the set of positive integers, yield useful formulas of a standard but not yet observed type.
scientific article; zbMATH DE number 1021687