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

• Main Authors: Peca-Medlin, John, Zhong, Chenyang
• Format: Journal Article
• Language: English
• Published: 28.10.2024
• Subjects: Computer Science - Numerical Analysis; Mathematics - Combinatorics; Mathematics - Numerical Analysis; Mathematics - Probability
• DOI: 10.48550/arxiv.2410.20952

For a square matrix $A$, Gaussian elimination with partial pivoting (GEPP) results in the factorization $PA = LU$ where $L$ and $U$ are lower and upper triangular matrices and $P$ is a permutation matrix. If $A$ is a random matrix, then the associated permutation from the $P$ factor is random. We are interested in studying properties of random permutations generated using GEPP. We introduce and present probabilistic results for groups of simple and nonsimple butterfly permutations of order $N = p^n$ for prime $p$, on which GEPP induces the Haar measure when applied to particular ensembles of random butterfly matrices. Of note, the nonsimple butterfly permutations include specific $p$-Sylow subgroups of $S_{p^n}$ for each prime $p$. Moreover, we focus on addressing the questions of the longest increasing subsequence (LIS) and the number of cycles for butterfly permutations. Our proof techniques utilize the explicit structural recursive properties of these permutations to answer what are traditionally hard analytical questions using simpler tools. For simple butterfly permutations, we provide full distributional descriptions and Law of Large Numbers (LLN) or Central Limit Theorem (CLT) type results for each statistic. For nonsimple $p$-nary butterfly permutations, we establish lower and upper power law bounds on the expected LIS of the form $N^{\alpha_p}$ and $N^{\beta_p}$ where $\frac12 < \alpha_p < \beta_p < 1$ for each $p$ with $\alpha_p = 1 - o_p(1)$. Additionally, we determine the aysmptotic form for the number of cycles scaled by $(2 - \frac1p)^n$ that limit to new distributions we introduce depending only on $p$ whose positive integer moments satisfy explicit recursion formulas for each $p$; in particular, this characterizes a full CLT result for the number of cycles for any uniform $p$-Sylow subgroup of $S_{p^n}$.