Notions of Möbius inversion

Möbius inversion formula for monoids with zero

The Möbius inversion formula, introduced during the 19th century in number theory, was generalized to a wide class of monoids called locally finite such as the free partially commutative, plactic and hypoplactic monoids for instance. In this contribution are developed and used some topological and algebraic notions for monoids with zero, similar to ordinary objects such as the (total) algebra o...

Counting Restricted Homomorphisms via Möbius Inversion over Matroid Lattices

We present a framework for the complexity classification of parameterized counting problems that can be formulated as the summation over the numbers of homomorphisms from small pattern graphs H1, . . . ,H` to a big host graph G with the restriction that the coefficients correspond to evaluations of the Möbius function over the lattice of a graphic matroid. This generalizes the idea of Curticape...

Fast Möbius Inversion in Semimodular Lattices and ER-labelable Posets

We consider the problem of fast zeta and Möbius transforms in finite posets, particularly in lattices. It has previously been shown that for a certain family of lattices, zeta and Möbius transforms can be computed in O(e) elementary arithmetic operations, where e denotes the size of the covering relation. We show that this family is exactly that of geometric lattices. We also extend the algorit...

The Calculation of Fourier Coefficients by the Möbius Inversion of the Poisson Summation Formula

In Part I, the MIPS method for calculating Fourier coefficients was introduced, and applied to functions / £ C(p>[0, 1]. In this part two extensions of the theory are described. One modification extends the theory to piecewise continuous functions, / £ PC(p'[0, 1]. Using these results the method may be used to calculate approximations to trigonometrical integrals (in which the length of the int...

Kant's Four Notions of Freedom