  We describe an infinite family of non-Pl\"ucker cluster variables inside the
double Bruhat cell cluster algebra defined by Berenstein, Fomin, and
Zelevinsky. These cluster variables occur in a family of subalgebras of the
double Bruhat cell cluster algebra which we call Double Rim Hook (DRH) cluster
algebras. We discover that all of the cluster variables are determinants of
matrices of special form. We conjecture that all the cluster variables of the
double Bruhat-cell cluster algebra have similar determinant form. We notice the
resemblance between our staircase diagram and Auslander-Reiten quivers.
