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Asymptotic normality of pattern counts in conjugacy classes

Valentin Féray, Mohamed Slim Kammoun

Abstract

We prove, under mild conditions on fixed points and two cycles, the asymptotic normality of vincular pattern counts for a permutation chosen uniformly at random in a conjugacy class.Additionally, we prove that the limiting variance is always non-degenerate for classical pattern counts. The proof uses weighted dependency graphs.

Asymptotic normality of pattern counts in conjugacy classes

Abstract

We prove, under mild conditions on fixed points and two cycles, the asymptotic normality of vincular pattern counts for a permutation chosen uniformly at random in a conjugacy class.Additionally, we prove that the limiting variance is always non-degenerate for classical pattern counts. The proof uses weighted dependency graphs.
Paper Structure (18 sections, 19 theorems, 89 equations, 3 figures)

This paper contains 18 sections, 19 theorems, 89 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Vincular patterns
  4. Our main result: asymptotic normality of vincular pattern counts for uniform random permutation in conjugacy classes
  5. Non-degeneracy of the limiting Gaussian law for classical patterns
  6. Preliminaries: weighted dependency graphs
  7. Definition
  8. A useful example: uniform random permutations in $S_n$
  9. Cumulant bounds
  10. Power of weighted dependency graphs
  11. A weighted dependency graph structure for uniform random permutation in conjugacy classes
  12. Proof of asymptotic normality of vincular pattern counts
  13. Non-degeneracy of the limiting law
  14. Preliminary: one point conditional pattern densities
  15. A recursive inequality on the variance
  16. ...and 3 more sections

Key Result

Theorem 1

For any vincular pattern $(\pi,A)$, there exist two polynomial functions $f$ and $g$ such that if $\frac{m_1(\lambda^{n})}{n} \to p_1$ and $\frac{m_2(\lambda^{n})}{n} \to p_2$, then Moreover, we have convergence of all moments.

Figures (3)

  • Figure 1: Example of a weighted graph $H$ with a marked spanning tree of maximum weight in red.
  • Figure 2: From left to right: the three graphs $L_1[\alpha]$, $K_1[\alpha]$ and $(S(\alpha),\alpha)$ associated with the $r=3$, $i_1=j_2=7$, $j_1=4$, $i_2=i_3=j_3=9$. Edges of the first kind in $K_1[\alpha]$ are plotted in blue. In this case, all three graphs are connected.
  • Figure 3: From left to right: the four graphs $G^{(1)}_{k_1,\ldots,k_r}$, $G^{(2)}_{k_1,\ldots,k_r}$, $G^\vee_{k_1,\ldots,k_r}$ and $G^\wedge_{k_1,\ldots,k_r}$ associated with the particular values of $i_1,\dots,i_r,j_1,\dots,j_r,k_1,\dots,k_r$ given on page \ref{['example']}.

Theorems & Definitions (33)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • Definition 2
  • Proposition 3
  • Lemma 4
  • Proposition 5
  • Lemma 6
  • proof
  • Theorem 3
  • ...and 23 more