In the last decade, the order polytope of the zigzag poset has been thoroughly studied. A related poset, called crown poset, obtained by adding an extra cover relation between the endpoints of an even zigzag poset, is not so well understood. In this paper, we study the order polytopes of crown posets. We provide explicit formulas for their f-vectors. We provide recursive formulas for their Ehrhart polynomial, giving a counterpart to formulas found in the zigzag case by Petersen and Zhuang (2025). We use these formulas to simplify a computation by Ferroni, Morales and Panova (2025) of the linear term of the order polynomial of these posets. Furthermore, we provide a combinatorial interpretation for the coefficients of the h & lowast;-polynomial in terms of the cyclic swap statistic on cyclically alternating permutations, which provides a circular version of a result by Coons and Sullivant (2023).

We prove that every lower unitriangular and totally nonnegative matrix gives rise to a family of polynomials with only real zeros. This has consequences for problems in several areas of mathematics. We use it to develop a general theory for chain enumeration in posets and zeros of chain polynomials. The results obtained extend and unify results of the first author, Brenti, Welker and Athanasiadis. In the process we define a notion of h -vector for a large class of posets which generalize the notions of h -vectors associated to simplicial and cubical complexes. A consequence of our methods is a characterization of the convex hull of all characteristic polynomials of hyperplane arrangements of fixed dimension and over a fixed finite field. This may be viewed as a refinement of the Critical Problem of Crapo and Rota. We also use the methods developed to solve an open problem posed by Forgács and Tran on the real-rootedness of polynomials arising from certain bivariate rational functions.