Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Major Index Distribution

Michael Coopman

TL;DR

This work introduces Maj(n,q), a distribution on the symmetric group $S_n$ with probability weight $q^{\mathrm{maj}(\pi)}$, and provides a transparent sampler built from $n$ i.i.d. Geometric$(1-q)$ variables via a map $\Gamma_n(G)$. It establishes limit-shape results through a permuton consisting of line segments with endpoints $(q^{i+1},0)$ to $(q^{i},1)$ and proves Brownian-type fluctuations, while also proving pattern counts are asymptotically normal and fixed-point statistics deviate from Poisson. The paper connects to $q$-Plancherel measure, recovering asymptotics for the limiting shape and fluctuations (as in Feray–Méliot) without heavy representation theory, providing a self-contained probabilistic framework. It further discusses 2-cycles, offers a nontrivial non-Poisson fixed-point result, and outlines promising directions for generalizing the sampler and varying $q$ with $n$, with potential implications for a broader class of deformation measures.

Abstract

For $0<q<1$, let $Maj$ be the distribution on the symmetric group $S_n$ such that a permutation $π\in S_n$ is selected with probability proportional to $q^{maj(π)}$. The distribution has connections to $q$-Plancherel measure. We describe an algorithm that realizes $Maj$, and use it to prove known results of $q$-Plancherel measure without the need of representation theory. This sampler is transparent and elegant, allowing properties of $Maj$ about its limit shape, pattern normality, and cycle structure to be obtained.

Major Index Distribution

TL;DR

This work introduces Maj(n,q), a distribution on the symmetric group with probability weight , and provides a transparent sampler built from i.i.d. Geometric variables via a map . It establishes limit-shape results through a permuton consisting of line segments with endpoints to and proves Brownian-type fluctuations, while also proving pattern counts are asymptotically normal and fixed-point statistics deviate from Poisson. The paper connects to -Plancherel measure, recovering asymptotics for the limiting shape and fluctuations (as in Feray–Méliot) without heavy representation theory, providing a self-contained probabilistic framework. It further discusses 2-cycles, offers a nontrivial non-Poisson fixed-point result, and outlines promising directions for generalizing the sampler and varying with , with potential implications for a broader class of deformation measures.

Abstract

For , let be the distribution on the symmetric group such that a permutation is selected with probability proportional to . The distribution has connections to -Plancherel measure. We describe an algorithm that realizes , and use it to prove known results of -Plancherel measure without the need of representation theory. This sampler is transparent and elegant, allowing properties of about its limit shape, pattern normality, and cycle structure to be obtained.
Paper Structure (13 sections, 22 theorems, 66 equations, 2 figures)

This paper contains 13 sections, 22 theorems, 66 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main Results
  3. Background and notation
  4. Geometric random variables
  5. Lattice walks, Brownian motion, and Donsker's theorem
  6. Permutons
  7. Permutation patterns and U-statistics
  8. Maj Sampler
  9. Limit Behavior
  10. Pattern Normality
  11. Fixed points and 2-cycles
  12. q-Plancherel Measure
  13. Future directions

Key Result

Theorem 1.1

Let $G = (G_{1}, \hdots, G_{n})$ be a sequence of i.i.d. geometric random variables with parameter $1-q$. Then, $\Gamma(G)$ follows the major index distribution.

Figures (2)

  • Figure 1: Two typical permutations taken from $\mathrm{Maj}(n,q)$. The left picture is for $\mathrm{Maj}(10000,0.5)$. The right picture is for $\mathrm{Maj}(10000,0.9)$.
  • Figure 2: The construction of the infinite sequence of lattice paths. Lattice walks can intersect but are drawn to look non-intersecting for clarity. The $G_{y}'s$ encode which lattice path gets an upright step at that height.

Theorems & Definitions (39)

  • Example
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: Theorem 2 of feray2012asymptotics
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 2.1: Memorylessness
  • Corollary 2.2: Memorylessness II
  • Proposition 2.3: Geometric races
  • proof
  • ...and 29 more