  In this paper, we compute the distributions of the statistic number of
crossings over permutations avoiding one of the pairs $\{321,231\}$,
$\{123,132\}$ and $\{123,213\}$. The obtained results are new combinatorial
interpretations of two known triangles in terms of restricted permutations
statistic. For other pairs of patterns of length three, we find relationships
between the polynomial distributions of the crossings over permutations that
avoid the pairs containing the pattern 231 on the first hand and the pattern
312 on the other hand.
