Relate the infinite-series Möbius inversion to the finite sums over connections and linear subdigraphs

Establish an explicit relationship between Leinster’s convergent infinite-series expression for the Möbius inverse on finite metric spaces, given by μ(x,y) = Σ_{k≥0} Σ_{x=x_0 ≠ x_1 ≠ … ≠ x_k=y} (-1)^k ζ(x_0,x_1) … ζ(x_{k-1},x_k), and the finite combinatorial sums over connections and linear subdigraphs obtained via Cramer’s formula for ζ_t(x,y)=e^{-t d(x,y) }, specifically the expressions for μ_t(x,y) and det ζ_t in Proposition \ref{prop:det_and_moebius_metric_spaces}. Determine conditions or constructions under which these formulations coincide or can be transformed into one another.

Background

In the metric-space setting, the paper recalls an infinite series formula for the Möbius inverse μ(x,y) due to Leinster that can be justified when the corresponding series converges, often under scattering conditions. This infinite expansion underpins interpretations connecting magnitude to magnitude homology.

The paper’s main contribution provides a different, finite combinatorial expression for μ_t(x,y) and det ζ_t by leveraging Cramer’s formula and interpreting determinants and cofactors through weighted connections and linear subdigraphs in the associated digraph. Despite both being valid formulations in overlapping regimes, the authors note they currently lack an explicit link between the two approaches.