Almost-valuative invariants of connected split matroids: The cd-index

Abstract

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 $\mathcal{P}(M)$, a polynomial in two non-commutative variables that compactly encodes the number of face-flags $\mathcal{F}=\{{\sigma }_1\subset \cdots \subset {\sigma }_s\}$ with prescribed dimensions $\dim {\sigma }_i=d_i$. This generalizes recent work by Ferroni and Schröter on the $f$-vector of $\mathcal{P}(M)$, 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 ${\Lambda }_{k,n}^{r,h}$; 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.