Extended Nestohedra and their Face Numbers

TL;DR

This paper introduces extended nestohedra, explores their geometric and combinatorial properties, proves they are boundaries of simple polytopes, and verifies Gal's conjecture for a broad class of these polytopes.

Contribution

It extends the class of nestohedra to include extended nestohedra, characterizes their duals, and provides formulas for their face vectors, advancing understanding of their structure and combinatorics.

Findings

Extended nestohedra are boundaries of simple polytopes.

All flag extended nestohedra have nonnegative gamma-vectors, confirming Gal's conjecture.

The paper provides explicit formulas for face and gamma-vectors in terms of descent statistics.

Abstract

Nestohedra are a family of convex polytopes that includes permutohedra, associahedra, and graph associahedra. In this paper, we study an extension of such polytopes, called extended nestohedra. We show that these objects are indeed the boundaries of simple polytopes, answering a question of Lam and Pylyavskyy. We also study the duals of (extended) nestohedra, giving a complete characterization of isomorphisms (as simplicial complexes) between the duals of extended nestohedra and a partial characterization of isomorphisms between the duals of nestohedra and extended nestohedra. In addition, we give formulas for their $f$-, $h$-, and $\gamma$-vectors. This includes showing that the $f$-vectors of the extended nestohedron corresponding to a forest $F$ and the nestohedron corresponding to the line graph of $F$ are the same, as well as showing that all flag extended nestohedra have nonnegative $\gamma$-vectors, thus proving Gal's conjecture for a large class of flag simple polytopes. We also relate the $f$- and $h$-vectors of the nestohedra and extended nestohedra, as well as give explicit formulas for the $h$- and $\gamma$-vectors in terms of descent statistics for a certain class of flag extended nestohedra. Finally, we define a partial ordering on partial permutations that is a join semilattice quotient of the weak Bruhar order on the symmetric group, and such that any linear extension of the partial order provides a shelling of the dual of the stellohedron.