Almost-valuative invariants of connected split matroids: The cd-index
We derive a formula for matroid invariants $\Psi$ on a large family of matroids, provided that $\Psi$ is almost-valuative, namely, it satisfies a hyperplane-cut formula. Our primary application is to the cd-index $\Psi_{cd}$ of the base polytope $\mathscr{P}(M)$, a polynomial in two non-commutative variables that compactly encodes the number of face-flags $\mathcal{F} = \{\sigma_1 \subset \dots \subset \sigma_s \}$ with prescribed dimensions $\dim \sigma_i = d_i$. This generalizes recent work by Ferroni and Schr\"oter on the $f$-vector of $\mathscr{P}(M)$, yielding a formula that can be understood as a valuative part plus an error term that surprisingly depends only on modular pairs of cyclic flats. This enables computations requiring only the following data: the evaluations of $\Psi$ on hypersimplices $\Delta_{k,n}$ and cuspidal matroids $\Lambda^{r,h}_{k,n}$; and counts $\lambda(r,h)$ and $\mu(a,b;\alpha,\beta)$ of cyclic flats and modular pairs of cyclic flats in $M$, respectively, satisfying specific rank and cardinality conditions. We compute these for the cd-index, yielding explicit results for sparse paving matroids and rank-2 matroids.