Parametric Möbius inversion formulas
type: Item
instance of: scholarly article
title: Parametric Möbius inversion formulas
zbMATH Open document ID: 0878.06003
DOI: 10.1016/S0012-365X(96)00085-4
author: Beifang Chen
published in: Discrete Mathematics
publication date: 1998-01-04T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 06A07
Mathematics Subject Classification ID: 05A15
zbMATH DE Number: 1021687
zbMATH Keywords: interval
zbMATH Keywords: set Möbius function
zbMATH Keywords: inversion
zbMATH Keywords: locally finite posets
zbMATH Keywords: incidence algebra
reviewed by: J. Neggers
MaRDI profile type: Publication
cites work: Q4491754
cites work: Q3856819
cites work: On the foundations of combinatorial theory I. Theory of M�bius Functions
cites work: Q3748279
description: scientific article; zbMATH DE number 1021687