On Some Properties of Permutation Tableaux

We consider the relations between various permutation statistics and properties of permutation tableaux. We answer some of the open problems of Steingrímsson and Williams [8], in particular, on the distribution of the bistatistic of numbers of rows and essential ones in permutation tableaux. We also consider and enumerate sets of permutation tableaux related to some pattern restrictions on permutations. Research supported in part by the NSA Young Investigator Grant H98230-06-1-0037.