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

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).