Juan Pablo Vigneaux Ariztia - A combinatorial approach to Möbius inversion and pseudoinversion

Abstract

Firstly,  I’ll explain how Cramer’s formula for the inverse of a matrix and a combinatorial expression for determinants give a novel combinatorial interpretation of the Möbius inverse whenever it exists. The sums thus obtained are indexed by linear connections on associated digraphs (each given by a path without cycles and a cycle decomposition of its complement). My result contains, as particular cases, previous theorems by P. Hall and T. Leinster.

Secondly, I’ll present a new definition of magnitude (valid for any finite category) in terms of the pseudo-Möbius function (Moore-Penrose pseudoinverse of the Dirichlet zeta matrix). This definition has been recently introduced by Akkaya and Ünlü and, independently, by Chen and me.  I’ll show how Berg’s formula for the computation of the Moore-Penrose pseudoinverse, the natural generalization of Cramer’s rule, yields a combinatorial interpretation of the pseudo-Möbius function, and therefore also of magnitude in all cases. This combinatorial interpretation a priori differs from the one given by magnitude homology.

Mangitude

The zeta function give the number of homsets. This requires the weightings to work out nicely.

Theorem for poset Let $(P,\le )$ be a poset ${\zeta }^{-1}(a,b)=\sum _{k\ge 0}(-1{)}^k\#\{\text{nondegenerate paths of lenght k from a to b}\}$ Threom (Leinster) The möbius inversion is $\mu (a,b)=\sum _{\text{paths a=c0→...ck=b points distinct}}$ In analogy to the matrix pseoduinverse ${\zeta }^{+}$ we introduce the pseudoinverse of the möbius inversion. This satisfieds the same relations.

Theorem henever $A$ has magnitude $\chi (A)=\sum _{a,b\in Ob(A)}{\zeta }^{+}(a,b)$ this extends the definition of magnitude to all finite categories.

If $A$ hat Möbius Inversion then $\mu (a,b)=\frac{1}{\det \zeta }\sum _{C\in \mathcal{C}(a\to b;D(A))}(-1{)}^{n+c(C)+1}w(C)$

Combinatorial linear algebra

Pseudoinversion

Comparison with previous results