Bott manifolds of Bott--Samelson type and assemblies of ordered partitions

A Bott manifold is a smooth projective toric variety having an iterated $\mathbb{C}{P}^1$-bundle structure. A certain family of Bott manifolds is used to understand the structure of Bott–Samelson varieties (or Bott–Samelson–Demazure–Hansen varieties), which provide desingularizations of Schubert varieties. Indeed, each Bott–Samelson variety is diffeomorphic to a Bott manifold. However, not all Bott manifolds originate from Bott–Samelson varieties. Those that do are specifically referred to as Bott manifolds of Bott–Samelson type. In this paper, we provide a characterization of Bott manifolds of Bott–Samelson type by exploring their relationship with combinatorial objects called assemblies of ordered partitions. Using this relationship, we enumerate Bott manifolds of Bott–Samelson type and describe isomorphic Bott manifolds of Bott–Samelson type in terms of assemblies of ordered partitions.