Weighted quasisymmetric enumerator for generalized permutohedra

Abstract

We introduce a weighted quasisymmetric enumerator function associated to generalized permutohedra. It refines the Billera, Jia and Reiner quasisymmetric function which also includes the Stanley chromatic symmetric function. Beside that it carries information of face numbers of generalized permutohedra. We consider more systematically the cases of nestohedra and matroid base polytopes.

Weighted P-partitions enumerator

“…In section 4 we prove Theorem 21, the first main result of the paper, which states that the weighted quasisymmetric function F q (C(P)), constructed geometrically, has an algebraic meaning as the universal morphism from a certain combinatorial Hopf algebra of posets P to the Hopf algebra QSym of quasisymmetric functions. This result is analogous to the previous results for simple graphs [7], matroids and building sets [8], and spreads their validity to the case of extended generalized permutohedra. The main theorem is followed by various examples, and statements about behavior of the enumerator of the opposite poset and under the action of the antipode.…”

“…The next theorem extends the similar statement proven for generalized permutohedra in [8] to the case of poset cones. Proof.…”

“…It was defined, and studied in the case of matroid base polytopes, by Billera, Jia and Reiner in [3], and in the case of nestohedra by Grujić in [6]. More subtle generalization which takes into account the face structure of a generalized permutohedron is introduced and studied in [8]. In this paper we consider the extended generalized permutohedra and the special case of poset cones.…”

“…In section 2 we review necessary facts about quasisymmetric functions and P-partitions enumerators. In section 3 we introduce the integer points enumerator F q (P ) for an extended generalized permutohedron P , which is a weighted quasisymmetric function, in the same manner as in the case of generalized permutohedra provided in [8]. The parameter q reflects the rank function of the face lattice L(P ).…”

Integer points enumerator of hypergraphic polytopes

“…This class of polytopes introduced by Postnikov [11] is distinguished with rich combinatorial structure. The comprehensive treatment of weighted integer points enumerators associated to generalized permutohedra is carried out by Grujić et al [8]. Here we consider a certain naturally defined non-cocommutative combinatorial Hopf algebra of hypergraphs and show that the derived quasisymmetric function invariant of hypergraphs is integer points enumerator of hypergraphic polytopes (Theorem 4.1).…”