j_2, \ldots, j_p) \in A_p$. We say $j_k, j_\ell$ is [*opposite sign pair matched*]{}, if $\epsilon_k$ and $\epsilon_\ell$ corresponding to $j_k$ and $j_\ell$, respectively, are of opposite sign and $j_k= j_\ell$, where $\epsilon_k$ and $\epsilon_\ell$ are corresponds to (\[def:A\_p\_SC\]). For example; In $(2,3,5,2)$, entry $2$ is [*opposite sign pair matched*]{}, if $\epsilon_1=1$ and $\epsilon_4=-1$ or $\epsilon_1=-1$ and $\epsilon_4=1$ whereas if $\epsilon_1$ = $\epsilon_4= 1$ or $\epsilon_1$ = $\epsilon_4= -1$, then $2$ is not [*opposite sign pair matched*]{}. Similarly, we can also define [*opposite sign pair matched*]{} elements of $\tilde{A}_p$. We shall call, vector $(j_1, j_2, \ldots, j_p)$ is [*opposite sign pair matched*]{}, if all the entries of $(j_1, j_2, \ldots, j_p)$ are [*opposite sign pair matched*]{}.

Observe that, if $(j_1, j_2, \ldots, j_p)\in A_{p}$, that is, $\sum_{i=1}^{p}\epsilon_i j_i=0 \mbox{ (mod $n$) }$ and each entry of $\{j_1, j_2, \ldots, j_p\}$ has multiplicity greater than or equal to two. Then the maximum number of free variable in $(j_1, j_2, \ldots, j_p)$ will be $\frac{p}{2}$ only when $p$ is even and $(j_1, j_2, \ldots, j_p)$ is [*opposite sign pair matched*]{}. We shall use this observation in proof of Theorem \[thm:symcircovar\], for maximum contribution. Now assuming the above Result, we proceed to prove 