Extensions of partial cyclic orders and consecutive coordinate polytopes

Abstract

We introduce several classes of polytopes contained in $[0,1]^n$ and cut out by inequalities involving sums of consecutive coordinates, extending a construction by Stanley. We show that the normalized volumes of these polytopes enumerate the extensions to total cyclic orders of certains classes of partial cyclic orders. We also provide a combinatorial interpretation of the Ehrhart $h^*$-polynomials of some of these polytopes in terms of descents of total cyclic orders. The Euler numbers, the Eulerian numbers and the Narayana numbers appear as special cases.