The probability of avoiding consecutive patterns in the Mallows distribution

We use various combinatorial and probabilistic techniques to study growth rates for the probability that a random permutation from the Mallows distribution avoids consecutive patterns. The Mallows distribution behaves like a $q$-analogue of the uniform distribution by weighting each permutation $\pi$ by $q^{inv(\pi)}$, where $inv(\pi)$ is the number of inversions in $\pi$ and $q$ is a positive, real-valued parameter. We prove that the growth rate exists for all patterns and all $q>0$, and we generalize Goulden and Jackson's cluster method to keep track of the number of inversions in permutations avoiding a given consecutive pattern. Using singularity analysis, we approximate the growth rates for length-3 patterns, monotone patterns, and non-overlapping patterns starting with 1, and we compare growth rates between different patterns. We also use Stein's method to show that, under certain assumptions on $q$, the length of $\sigma$, and $inv(\sigma)$, the number of occurrences of a given pattern $\sigma$ is well approximated by the normal distribution.