Extensions of partial cyclic orders and consecutive coordinate polytopes

Arvind Ayyer (1) , Matthieu Josuat-Vergès (2, 3) , Sanjay Ramassamy (4)

1 Department of Mathematics, Indian Institute of Science
2 LIGM - Laboratoire d'Informatique Gaspard-Monge
3 CNRS - Centre National de la Recherche Scientifique
4 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

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.

Domains: Mathematics [math]; Combinatorics [math.CO]

Arvind Ayyer, Matthieu Josuat-Vergès, Sanjay Ramassamy. Extensions of partial cyclic orders and consecutive coordinate polytopes. Annales Henri Lebesgue, 2020, 3, pp.275-297. ⟨10.5802/ahl.33⟩. ⟨ensl-01745941v2⟩