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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 unionsof cycles in the rectangular grid graph $P_m \times P_n$ ($m,n \in N$), when$m$ is odd, is equal to $\displaystyle \mid V({\cal R}^{**}_{m}) \mid =\frac{1}{2} \left[{m+1 \choose (m-1)/2 } + {(m+1)/2 \choose \lfloor (m+1)/4\rfloor}\right].$

A short note on the order of the double reduced 2-factor transfer digraph for rectangular grid graphs