  Some equivalence classes in symmetric group lead to an interesting class of
noncommutive and associative algebras. From these algebras we construct
noncommutative Frobenius algebras. Based on the correspondence between
Frobenius algebras and two-dimensional topological field theories, the
noncommutative Frobenius algebras can be interpreted as topological open string
theories. It is observed that the centre of the algebras are related to closed
string theories via open-closed duality. Area-dependent two-dimensional field
theories are also studied.
