  In this paper, we show that a $D$-type map $\Phi_D:M_n\rightarrow M_n$ with
$D=(n-2)I_n+P_{\pi_1}+P_{\pi_2}$ induced by a pair
  $\{\pi_1,\pi_2\}$ of permutations of $(1,2,..., n)$ is positive if
$\{\pi_1,\pi_2\}$ has property (C). The property (C) is characterized for
$\{\pi_1,\pi_2\}$, and an easy criterion is given for the case that
$\pi_1=\pi^p$ and $\pi_2=\pi^q$, where $\pi$ is the permutation defined by
$\pi(i)=i+1$ mod $n$ and $1\leq p<q\leq n$.
