  Let $I$ be a regular $\mathfrak m$-primary ideal in $(R,\mathfrak m,k)$. Then
the Ratliff-Rush ideal associated to $I$ is denoted by $\bar I$ and is defined
as the largest ideal containing $I$ with the same Hilbert polynomial as $I$. In
this paper we present a method to compute Ratliff-Rush ideals for a certain
class of monomial ideals in the rings $k[x,y]$ and $k[[x,y]]$. We find an upper
bound for the Ratliff-Rush reduction number for an ideal in this class.
Moreover, we establish some new characterizations of when all powers of $I$ are
Ratliff-Rush.
