  Let $E=((e_{ij}))_{n\times n}$ be a fixed array of real numbers such that
$e_{ij}=e_{ji}, e_{ii}=0$ for $1\le i,j \le n$. Let the permutation group be
denoted by $S_n$ and the collection of involutions with no fixed points by
$\Pi_n$, that is, $\Pi_n=\{\pi\in S_n: \pi^2= id, \pi(i)\neq i\,\forall i\}$
with id denoting the identity permutation. For $\pi$ uniformly chosen from
$\Pi_n$, let $Y_E=\sum_{i=1}^n e_{i\pi(i)}$ and $W=(Y_E-\mu_E)/\sigma_E$ where
$\mu_E=E(Y_E)$ and $\sigma_E^2= Var(Y_E)$. Denoting by $F_W$ and $\Phi$ the
distribution functions of $W$ and a $\mathcal{N}(0,1)$ variate respectively, we
bound $||F_W-\Phi||_p$ for $ 1\le p\le \infty$ using Stein's method and the
zero bias transformation. Optimal Berry-Esseen or $L^\infty$ bounds for the
classical problem where $\pi$ is chosen uniformly from $S_n$ were obtained by
Bolthausen using Stein's method. Although in our case $\pi \in \Pi_n$
uniformly, the $L^p$ bounds we obtain are of similar form as Bolthausen's bound
which holds for $p=\infty$. The difficulty in extending Bolthausen's method
from $S_n$ to $\Pi_n$ arising due to the involution restriction is tackled by
the use of zero bias transformations.
