The $h^*$-polynomials of type C hypersimplices

Abstract: We study the Ehrhart theory of hypersimplices of type C, as introduced by Lam and Postnikov for general crystallographic root systems. The $h^*$-polynomials of classical hypersimplices are known to relate to various Eulerian statistics on the symmetric group. In this paper, we introduce a new statistic and partial order on signed permutations, which we use to derive explicit formulas for the $h^*$-polynomials of type C hypersimplices. Additionally, we explore connections with other statistics, including flag-excedances and circular descents, flag-descents, and Coxeter descents.