The Combinatorics of Cyclic Orders

Abstract: A cyclic order on a set X is a ternary relation on X satisfying three conditions called cyclicity, asymmetry and transitivity. These can be thought of as analogues of partial orders. There are many similarities between posets and cyclically ordered sets. In particular, there is a notion of a circular extension and there is a natural generalisations of the order polytope defined by Stanley. Cyclic orders were recently introduced in the combinatorial literature to enumerate descent classes of permutations.

We introduce several classes of polytopes contained in $[0,1]^n$ and cut out by inequalities involving sums of consecutive coordinates. We show that the normalized volumes of these polytopes enumerate circular extensions of certain partial cyclic orders. Among other things this gives a new point of view on a question popularized by Stanley (Exercise~4.56(d) in Enumerative combinatorics, Volume 1). 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 (a joint work with M. Josuat-Verg\`es and S. Ramassamy).