  We prove that the order of the double reduced 2-factor transfer digraph
${\cal R}^{**}_{m}$ which is needed for the enumeration of the spanning unions
of cycles in the rectangular grid graph $P_m \times P_n$ ($m,n \in N$), when
$m$ is odd, is equal to $\displaystyle \mid V({\cal R}^{**}_{m}) \mid =
\frac{1}{2} \left[{m+1 \choose (m-1)/2 } + {(m+1)/2 \choose \lfloor (m+1)/4
\rfloor}\right].$
