Discrete Faà di Bruno via Möbius Inversion

We approach discrete and differential Fa\`a di Bruno formulas from a M\"obius inversion angle. On the Boolean cube, Newton's discrete Taylor formula and the definition of iterated forward differences form a zeta--M\"obius dual pair, and composing two Taylor expansions and inverting once yields a closed discrete Fa\`a di Bruno formula at a fixed basepoint: for arbitrary maps $f, g$ between abelian groups, $$\Delta(f \circ g;\,x;\,u_1,\dots,u_k) = \sum_{H \in \mathrm{Cov}(k)} \Delta(f;\,g(x);\,(\Delta(g;x;u_T))_{T\in H}),$$ where $\mathrm{Cov}(k)$ denotes the coverings of $[k]$ by nonempty subsets. Grouping repeated directions gives binomial versions on multi-index grids, and iterating gives formulas for $m$-fold composites, with integer covering coefficients governed by explicit cross and level recursions, a discrete analogue of the Constantine--Savits formulas. The relationship between coverings and partitions appearing in classical Fa\`a di Bruno formulas is exhibited in an algebraic setting. The discrete formulas are Taylor expansions over the function algebra of the Boolean cube, whose idempotent generators absorb overlapping products; in the differential analogue nilpotent generators annihilate overlaps and only partitions remain. We demonstrate how these algebraic identities can be lifted to the analytical setting of $C^n$ maps between Banach spaces, recovering the multivariate Fa\`a di Bruno formula of Constantine--Savits and extending it to composites of several maps. Boolean finite differences, binomial grid formulas, infinitesimal Taylor algebras, and Fr\'echet derivatives thus appear as four realizations of one M\"obius-dual Fa\`a di Bruno formula, connected by a flat family.