  We study a system of monodispersed hard rectangles of size $m \times d$,
where $d\geq m$ on a two dimensional square lattice. For large enough aspect
ratio, the system is known to undergo three entropy driven phase transitions
with increasing activity $z$: first from disordered to nematic, second from
nematic to columnar and third from columnar to sublattice phases. We study the
nematic-columnar transition by developing a high-activity expansion in integer
powers of $z^{-1/d}$ for the columnar phase in a model where the rectangles are
allowed to orient only in one direction. By deriving the exact expression for
the first $d+2$ terms in the expansion, we obtain lower bounds for the critical
density and activity. For $m$, $k\gg 1$, these bounds decrease with increasing
$k$ and decreasing $m$.
