Symmetry properties of gamma matrices While reading a paper on supersymmetry i faced the following problem. Its about the symmetry property of charge conjugation matrix in different space time dimension. The charge conjugation matrix is defined as
\begin{equation}
C^{T} = -\epsilon C
\end{equation}
Problem is to find out the $\epsilon$ as a function of space-time dimension. 
If the spacetime dimension $\mathit{D}$ is even, the Clifford generators are given by
\begin{equation}
I, \gamma_{m}, \gamma_{m_{1}m_{2}},\gamma_{{m_{1}m_{2}}m_{3}},\dots\gamma_{{m_{1}m_{2}}m_{3}\dots m_{D}}
\end{equation}
There are altogether $2^D$ such matrices.
Following little algebra it can be shown that
\begin{equation}
(C\gamma_{{m_{1}m_{2}}m_{3}\dots m_{D}})^{T} = \epsilon (-1)^{\frac{(p-1)(p-2)}{2}} C\gamma_{{m_{1}m_{2}}m_{3}\dots m_{D}}
\end{equation}
Which means $C\gamma_{{m_{1}m_{2}}m_{3}\dots m_{D}}$ either symmetric or anti symmetric.
It says number of anti symmetric matrices in this set is equal to
\begin{equation}
\sum_{p=0}^{D}\frac{1}{2}(1-\epsilon(-1)^{\frac{(p-1)(p-2)}{2}})\binom{D}{p}
\end{equation} 
Can anyone help to derive the last expression? It will be a great help. Thanks in advance.
 A: In even dimension $D$, there are $\begin{pmatrix}D\\r\end{pmatrix}$ generators of rank $r$, $\gamma_{\mu_1 .. \mu_r}$. This is because $\gamma_{\mu_1 .. \mu_r}=\gamma_{[\mu_1}..\gamma_{\mu_r]}$ where $[..]$ denotes antisymmetrization. The index $\mu_i$ rund from $1$ to $D$. Then consider how many independent rank $r$ object you can write down, considering that using the same index twice will give you $0$ under antisymmetrization. All generators of the same rank have the same symmetry under $(C \gamma_{\mu_1 .. \mu_r})^T$. Take $t_r=1$ for rank $r$ with antisymmetric generators and $t_r=-1$ for rank with symmetric generators. Then clearly $\frac{1}{2}(1+t_r)$ is $0$ for symmetric ranks and $1$ for antisymmetric ranks. Then the total number of antisymmetric generators is
\begin{equation}
\sum_{r=0}^D \frac{1}{2}(1+t_r)\begin{pmatrix}D\\r\end{pmatrix}
\end{equation}
which is already quite close to what you desire. Now, I assume that where you wrote $(C \gamma_{m_1..m_D})^T$, you meant $(C \gamma_{m_1..m_p})^T$. Then using your expressions, we get $t_r = -\epsilon(-1)^{(r-1)(r-2)/2}$. Filling this in, we get the expression you desire.
For more information, see chapter 3 of Supergravity by Freedman & Van Proeyen, especially sections 3.1.2 , 3.1.8 and 3A.4 .
