Combinatorial Properties of Permutation Tableaux

We give another construction of a permutation tableau from its corresponding permutation and construct a permutation-preserving bijection between 1-hinge and 0-hinge tableaux. We also consider certain alignment and crossing statistics on permutation tableaux that are known to be equidistributed via a complicated map to permutations that translates those to occurrences of certain patterns. We give two direct maps on tableaux that proves the equidistribution of those statistics by exchanging some statistics and preserving the rest. Finally, we enumerate some sets of permutations that are restricted both by pattern avoidance and by certain parameters of their associated permutation tableaux