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In this paper, we compute the distributions of the statistic number ofcrossings over permutations avoiding one of the pairs $\{321,231\}$,$\{123,132\}$ and $\{123,213\}$. The obtained results are new combinatorialinterpretations of two known triangles in terms of restricted permutationsstatistic. For other pairs of patterns of length three, we find relationshipsbetween the polynomial distributions of the crossings over permutations thatavoid the pairs containing the pattern 231 on the first hand and the pattern312 on the other hand.

Crossings over permutations avoiding some pairs of patterns of length three