M\"obius Polynomials of Face Posets of Convex Polytopes

TL;DR

This paper investigates the M"obius polynomial of face posets of convex polytopes, providing formulas for various polytope constructions and classes, enhancing understanding of their combinatorial invariants.

Contribution

It introduces new formulas for computing M"obius polynomials of face posets of convex polytopes, including pyramids, prisms, and glued polytopes, and relates them to $f$-vectors and Eulerian properties.

Findings

Formulas for pyramids and prisms M"obius polynomials

Methods for glued polytopes M"obius polynomial calculation

Expressions for simplicial and Eulerian polytopes

Abstract

The M\"obius polynomial is an invariant of ranked posets, closely related to the M\"obius function. In this paper, we study the M\"obius polynomial of face posets of convex polytopes. We present formulas for computing the M\"obius polynomial of the face poset of a pyramid or a prism over an existing polytope, or of the gluing of two or more polytopes in terms of the M\"obius polynomials of the original polytopes. We also present general formulas for calculating M\"obius polynomials of face posets of simplicial polytopes and of Eulerian posets in terms of their $f$-vectors and some additional constraints.