On the longest increasing subsequence and number of cycles of butterfly permutations

Abstract

One method to generate random permutations involves using Gaussian elimination with partial pivoting (GEPP) on a random matrix $A$ and storing the permutation matrix factor $P$ from the resulting GEPP factorization $PA=LU$. We are interested in exploring properties of random butterfly permutations, which are generated using GEPP on specific random butterfly matrices. Our paper highlights new connections among random matrix theory, numerical linear algebra, group actions of rooted trees, and random permutations. We address the questions of the longest increasing subsequence (LIS) and number of cycles for particular uniform butterfly permutations, with full distributional descriptions and limit theorems for simple butterfly permutations. We also establish scaling limit results and limit theorems for nonsimple butterfly permutations, which include certain $p$-Sylow subgroups of the symmetric group of $N=p^n$ elements for prime $p$. For the LIS, we establish power law bounds on the expected LIS of the form $N^{α_p}$ and $N^{β_p}$ where $\frac12 < α_p < β_p < 1$ for each $p$ with $α_p = 1 - o_p(1)$, showing distinction from the typical $O(N^{1/2})$ expected LIS frequently encountered in the study of random permutations (e.g., uniform permutations). For the number of cycles scaled by $(2-1/p)^n$, we establish a full CLT to a new limiting distribution depending on $p$ with positive support we introduce that is uniquely determined by its positive moments that satisfy explicit recursive formulas; this thus determines a CLT for the number of cycles for any uniform $p$-Sylow subgroup of $S_{p^n}$.