Permutation Tableaux and the Dashed Permutation Pattern 32-1

Abstract: We give a solution to a problem posed by Corteel and Nadeau concerning permutation tableaux of length n and the number of occurrences of the dashed pattern 32--1 in permutations on [n]. We introduce the inversion number of a permutation tableau. For a permutation tableau T and the permutation $\pi$ obtained from T by the bijection of Corteel and Nadeau, we show that the inversion number of T equals the number of occurrences of the dashed pattern 32--1 in the reverse complement of $\pi$. We also show that permutation tableaux without inversions coincide with L-Bell tableaux introduced by Corteel and Nadeau.