Zig-zag Eulerian polynomials

For any finite partially ordered set $P$, the $P$-Eulerian polynomial is the generating function for the descent number over the set of linear extensions of $P$, and is closely related to the order polynomial of $P$ arising in the theory of $P$-partitions. Here we study the $P$-Eulerian polynomial where $P$ is a naturally labeled zig-zag poset; we call these zig-zag Eulerian polynomials. A result of Br\"and\'en implies that these polynomials are gamma-nonnegative, and hence their coefficients are symmetric and unimodal. The zig-zag Eulerian polynomials and the associated order polynomials have appeared fleetingly in the literature in a wide variety of contexts$\unicode{x2014}$e.g., in the study of polytopes, magic labelings of graphs, and Kekul\'e structures$\unicode{x2014}$but they do not appear to have been studied systematically. In this paper, we use a "relaxed" version of $P$-partitions to both survey and unify results. Our technique shows that the zig-zag Eulerian polynomials also capture the distribution of "big returns" over the set of (up-down) alternating permutations, as first observed by Coons and Sullivant. We develop recurrences for refined versions of the relevant generating functions, which evoke similarities to recurrences for the classical Eulerian polynomials. We conclude with a literature survey and open questions.