Thresholds for Patterns in Random Permutations with a Given Number of Inversions - Bevan, David; Threlfall, Dan (The Electronic Journal of Combinatorics, 2024)
We explore how the asymptotic structure of a random permutation of $[n]$ with $m$ inversions evolves, as $m$ increases, establishing thresholds for the appearance and disappearance of any classical, consecutive or vincular pattern. The threshold for the appearance of a classical pattern depends on the greatest number of inversions in any of its sum indecomposable components.