Arithmetic of marked order polytopes, monotone triangle reciprocity, and partial colorings

For a poset P, a subposet A, and an order preserving map F from A into the real numbers, the marked order polytope parametrizes the order preserving extensions of F to P. We show that the function counting integral-valued extensions is a piecewise polynomial in F and we prove a reciprocity statement in terms of order-reversing maps. We apply our results to give a geometric proof of a combinatorial reciprocity for monotone triangles due to Fischer and Riegler (2011) and we consider the enumerative problem of counting extensions of partial graph colorings of Herzberg and Murty (2007).Comment: 17 pages, 10 figures; V2: minor changes (including title); V3: examples included (suggested by referee), to appear in "SIAM Journal on Discrete Mathematics

On f- and h- vectors of relative simplicial complexes

A relative simplicial complex is a collection of sets of the form $\Delta \setminus \Gamma$, where $\Gamma \subset \Delta$ are simplicial complexes. Relative complexes played key roles in recent advances in algebraic, geometric, and topological combinatorics but, in contrast to simplicial complexes, little is known about their general combinatorial structure. In this paper, we address a basic question in this direction and give a characterization of $f$-vectors of relative (multi)complexes on a ground set of fixed size. On the algebraic side, this yields a characterization of Hilbert functions of quotients of homogeneous ideals over polynomial rings with a fixed number of indeterminates. Moreover, we characterize $h$-vectors of fully Cohen--Macaulay relative complexes as well as $h$-vectors of Cohen--Macaulay relative complexes with minimal faces of given dimensions. The latter resolves a question of Bj\"orner.Comment: accepted for publication in Algebraic Combinatoric

Combinatorial mixed valuations

Combinatorial mixed valuations associated to translation-invariant valuations on polytopes are introduced. In contrast to the construction of mixed valuations via polarization, combinatorial mixed valuations reflect and often inherit properties of inhomogeneous valuations. In particular, it is shown that under mild assumptions combinatorial mixed valuations are monotone and hence nonnegative. For combinatorially positive valuations, this has strong computational implications. Applied to the discrete volume, the results generalize and strengthen work of Bihan (2015) on discrete mixed volumes. For rational polytopes, it is proved that combinatorial mixed monotonicity is equivalent to monotonicity. Stronger even, a conjecture is substantiated that combinatorial mixed monotonicity implies the homogeneous monotonicity in the sense of Bernig--Fu (2011).Comment: 17 pages, minor changes, accepted for publication in Adv. Mat