  We introduce the quasi-partition algebra $QP_k(n)$ as a centralizer algebra
of the symmetric group. This algebra is a subalgebra of the partition algebra
and inherits many similar combinatorial properties. We construct a basis for
$QP_k(n)$, give a formula for its dimension in terms of the Bell numbers, and
describe a set of generators for $QP_k(n)$ as a complex algebra. In addition,
we give the dimensions and indexing set of its irreducible representations. We
also provide the Bratteli diagram for the tower of quasi-partition algebras
(constructed by letting $k$ range over the positive integers).
