The type $B$ permutohedron and the poset of intervals as a Tchebyshev transform

TL;DR

This paper explores a Tchebyshev triangulation of the order complex of a poset's intervals and reveals a combinatorial equivalence between the dual of the type B permutohedron and a suspension of a Boolean algebra's interval poset.

Contribution

It introduces a new Tchebyshev triangulation of poset interval complexes and establishes a novel combinatorial equivalence involving the type B permutohedron.

Findings

Order complex of intervals forms a Tchebyshev triangulation

Dual of type B permutohedron is equivalent to a suspension of a Boolean algebra interval complex

Properties of the Tchebyshev transformation are analyzed

Abstract

We show that the order complex of intervals of a poset, ordered by inclusion, is a Tchebyshev triangulation of the order complex of the original poset. Besides studying the properties of this transformation, we show that the dual of the type $B$ permutohedron is combinatorially equivalent to the suspension of the order complex of the poset of intervals of a Boolean algebra (with the minimum and maximum elements removed).