  The quasi-partition algebras were introduced by Daugherty and the first
author as centralizers of the symmetric group. In this article, we give a more
general definition of these algebras and give a construction of their simple
modules. In addition, we introduce two new algebras, we give linear bases and
show that for specializations of their parameters, these new algebras are
isomorphic to centralizer algebras. We provide a generalized Bratteli diagram
that illustrates how the representation theory of the three algebras discussed
in this paper are related. Moreover, we give combinatorial formulas for the
dimensions of the simple modules of these algebras.
