Abstract. We generalize the definition of the [Formula: see text]-index of an Eulerian poset to the class of semi-Eulerian posets. For connected simplicial semi-Eulerian Buchsbaum posets, we show that all coefficients of the [Formula: see text]-index are nonnegative. This proves a conjecture of Novik for odd-dimensional manifolds and extends it to the even-dimensional case.

Abstract. We study [Formula: see text]-vectors associated with [Formula: see text]-vectors of symmetric edge polytopes both from a deterministic and a probabilistic point of view. On the deterministic side, we prove nonnegativity of [Formula: see text] for any graph and completely characterize the case when [Formula: see text]. The latter also confirms a conjecture by Lutz and Nevo in the realm of symmetric edge polytopes. On the probabilistic side, we show that the [Formula: see text]-vectors of symmetric edge polytopes of most Erdős–Rényi random graphs are asymptotically almost surely nonnegative up to any fixed entry. This proves that Gal’s conjecture holds asymptotically almost surely for arbitrary unimodular triangulations in this setting.

We investigate the question of whether any d-colorable simplicial d-polytope can be octahedralized, i.e., can be subdivided to a d-dimensional geometric cross-polytopal complex. We give a positive answer in dimension 3, with the additional property that the octahedralization introduces no new vertices on the boundary of the polytope.