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

We prove that the order of the double reduced 2-factor transfer digraph ${\mathcal{R}}_m^{\ast \ast }$ 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 ${\displaystyle \mid V({\mathcal{R}}_m^{\ast \ast })\mid =\frac{1}{2}[\left(\genfrac{}{}{0px}{}{m+1}{(m-1)\mathrm{/}2}\right)+\left(\genfrac{}{}{0px}{}{(m+1)\mathrm{/}2}{\lfloor (m+1)\mathrm{/}4\rfloor }\right)]\mathrm{.}}$