Asymptotic normality of statistics on permutation tableaux

We use a probabilistic approach to give new derivations of the expressions for the probability generating functions of basic statistics defined on permutation tableaux. Since our expressions are exact, we can identify the distributions of such basic statistics as the number of unrestricted rows, the number of rows, and the number of 1s in the first row. All three distributions were known, via involution on permutation tableaux and bijections between permutation tableaux and permutations, to be asymptotically normal after suitable normalizations. Our technique gives direct proofs of these results, as it allows us to work directly with permutation tableaux. We also establish the asymptotic normality of the number of superfluous 1s. This last result relies on a bijection between permutation tableaux and permutations, and on a rather general sufficient condition for the central limit theorem for the sums of random variables, in terms of a dependency graph of the summands.