Tommaso Faustini | Search arXiv

arXiv2026

Stellahedral Geometry of Partially Ordered Sets

We introduce a transformation on partially ordered sets, termed the \emph{stellahedral transform}, with notable features. It preserves the properties of being Eulerian, Cohen--Macaulay, and of being the face poset of a polytope. Furthermore, it admits an explicit geometric realization for convex polytopes and specializes to the construction that takes a simplex to the stellahedron. One motivation for this definition comes from the theory of toric $h$-polynomials and (augmented) Chow polynomials of Eulerian posets. We show that the right augmented Chow polynomial of an Eulerian poset $P$ agrees with the toric $h$-polynomial of the stellahedral transform of $P$. We use this perspective, together with $\mathbf{cd}$-index results due to Ehrenborg (2005) and Karu (2006), to prove two positivity results for augmented Chow polynomials: for Gorenstein* posets they are unimodal, and for face posets of polytopes they are $γ$-positive. Along the way we provide negative answers to two open questions concerning Eulerian and Gorenstein* posets. First, the question on the nonnegativity of Eulerian Chow polynomials, posed by Ferroni, Matherne, and Vecchi (2024). Second, the question posed by Athanasiadis and Kalampogia-Evangelinou (2023) on the real-rootedness of chain and Chow polynomials of Gorenstein* posets: these examples provide a novel application of a technique introduced by Murai and Nevo (2014).

math.CO↗

arXiv2026

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

We derive a formula for matroid invariants $Ψ$ on a large family of matroids, provided that $Ψ$ is almost-valuative, namely, it satisfies a hyperplane-cut formula. Our primary application is to the cd-index $Ψ_{cd}$ of the base polytope $\mathscr{P}(M)$, a polynomial in two non-commutative variables that compactly encodes the number of face-flags $\mathcal{F} = \{σ_1 \subset \dots \subset σ_s \}$ with prescribed dimensions $\dim σ_i = d_i$. This generalizes recent work by Ferroni and Schröter 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 $Ψ$ on hypersimplices $Δ_{k,n}$ and cuspidal matroids $Λ^{r,h}_{k,n}$; and counts $λ(r,h)$ and $μ(a,b;α,β)$ 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.

math.CO↗