Permutations avoiding a pattern of length three under Mallows distributions

Abstract

We consider permutations avoiding a pattern of length three under the family of Mallows distributions. In particular, for any pattern $\tau \in S_3-\{321\}$, we obtain rather precise results on the asymptotic probability as $n\to \infty$ that a permutation $\sigma \in S_n$ under the Mallows distribution with parameter $q\in (0,1)$ avoids the pattern. By a duality between the parameters $q$ and $\frac{1}{q}$, we also obtain rather precise results on the above probability for $q>1$ and any pattern $\tau \in S_3-\{123\}$.

There was an error in the proof of Theorem 2 which has been corrected in this version. Also, a memorial dedication has been added