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

John Peca-Medlin, Chenyang Zhong

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 $S_{p^n}$.

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 and storing the permutation matrix factor from the resulting GEPP factorization . 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 -Sylow subgroups of the symmetric group of elements for prime . For the LIS, we establish power law bounds on the expected LIS of the form and where for each with , showing distinction from the typical expected LIS frequently encountered in the study of random permutations (e.g., uniform permutations). For the number of cycles scaled by , we establish a full CLT to a new limiting distribution depending on 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 -Sylow subgroup of .

Paper Structure

This paper contains 23 sections, 39 theorems, 130 equations, 8 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Overview of results
  3. Notation and preliminaries
  4. Random permutations using GEPP
  5. Random butterfly matrices
  6. Simple butterfly permutations
  7. Nonsimple butterfly permutations
  8. n-nary butterfly permutations
  9. Diagonal butterfly permutations
  10. Longest increasing subsequence
  11. Simple butterfly permutations
  12. n
  13. General n
  14. Nonsimple butterfly permutations
  15. n
  16. ...and 8 more sections

Key Result

Theorem 1

If $i_k \sim \operatorname{Unif}(\{k,k+1,\ldots,n\})$ are independent for $k = 1,\ldots,n$, then $(n \ i_n) \cdots$$(2 \ i_2)(1 \ i_1) \sim \operatorname{Unif}(S_n)$.

Figures (8)

  • Figure 1: Diagrams for random permutations of size $N = 2^{10} = 1024$ for (a) uniform, (b) simple binary butterfly, (c) nonsimple binary butterfly, and (d) diagonal butterfly permutations, and $N = 3^7 = 2187$ for (e) simple ternary butterfly and (f) nonsimple ternary butterfly permutations.
  • Figure 2: Three views of the LIS $(4 \ 5 \ 6 \ 7)$ within $(4 \ 8 \ 5 \ 1 \ 3 \ 6 \ 7 \ 2)$, as seen in the (a) associated permutation matrix, (b) diagram of plotted pairs $(j,\sigma(j))/8 \subset [0,1]^2$, and (c) corresponding Young tableau where the length of the top row yields the LIS.
  • Figure 3: Diagrams for a LIS (in red) for random permutations of size $N = 2^{8} = 256$ for (a) uniform, (b) simple binary butterfly, (c) nonsimple binary butterfly, and (d) diagonal butterfly permutations, and $N = 3^5 = 243$ for (e) simple ternary butterfly and (f) nonsimple ternary butterfly permutations.
  • Figure 4: Standard and log-log plot of sample means to estimate $\mathbb{E} L(\sigma)$ for random permutations using GEPP, along with comparisons of $2 N^{1/2}$ for uniform permutations, $(3/2)^n$ for simple butterfly permutations, and exact computations of $\mathbb{E} L(\sigma)$ for nonsimple butterfly permutations, for $N = 2^n$ with $n = 1,2,\ldots,15$.
  • Figure 5: Comparing $\alpha_m$ to $\beta_m$.
  • ...and 3 more figures

Theorems & Definitions (59)

  • Theorem 1: FY38
  • Theorem 2: DiSh87
  • Theorem 3: P24
  • Corollary 1: P24
  • Theorem 4: P24
  • Proposition 1
  • proof
  • Corollary 2
  • Theorem 5
  • proof
  • ...and 49 more