Abstract
We use 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 is a q-analogue of the uniform distribution weighting each permutation π by qinv(π), where inv(π) is the number of inversions in π 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 and σ, the number of occurrences of a given pattern σ is well approximated by the normal distribution.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 417-447 |
| Number of pages | 31 |
| Journal | Random Structures and Algorithms |
| Volume | 53 |
| Issue number | 3 |
| DOIs | |
| State | Published - Oct 2018 |
All Science Journal Classification (ASJC) codes
- Software
- General Mathematics
- Computer Graphics and Computer-Aided Design
- Applied Mathematics
Keywords
- Mallows distribution
- Stein's method
- consecutive pattern
- inversion
- permutation