  Given two operators $\hat D$ and $\hat E$ subject to the relation $\hat D\hat
E -q \hat E \hat D =p$, and a word $w$ in $M$ and $N$, the rewriting of $w$ in
normal form is combinatorially described by rook placements in a Young diagram.
We give enumerative results about these rook placements, particularly in the
case where $p=(1-q)/q^2$. This case naturally arises in the context of the
PASEP, a random process whose partition function and stationary distribution
are expressed using two operators $D$ and $E$ subject to the relation
$DE-qED=D+E$ (matrix Ansatz). Using the link obtained by Corteel and Williams
between the PASEP, permutation tableaux and permutations, we prove a conjecture
of Corteel and Rubey about permutation enumeration. This result gives the
generating function for permutations of given size with respect to the number
of ascents and occurrences of the pattern 13-2, this is also the moments of the
$q$-Laguerre orthogonal polynomials.
