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Moments of permutation statistics by cycle type

Zachary Hamaker, Brendon Rhoades

TL;DR

The work develops a unified framework to analyze moments and asymptotics of permutation statistics conditioned on cycle type. By expressing statistics as linear combinations of partial permutation indicators and exploiting Möbius inversion, the authors prove a key polynomiality result: for a partial permutation of size $k$ with cycle‑path type $(\mu,\nu)$, $(n)_m \mathbb{E}_\lambda[1_{IJ}]$ equals a polynomial $f_{(\mu,\nu)}(n,m_1(\lambda),\dots,m_k(\lambda))$. They introduce regular statistics, show their moments are polynomials with explicit degree bounds, and demonstrate closure under multiplication, enabling a broad moment theory that includes classical and bivincular pattern counts. The paper derives asymptotic behavior where the mean and variance depend only on fixed points and two‑cycles, establishes LLNs and permuton limits, and discusses extensions to other groups and combinatorial objects. This framework not only generalizes prior results (Zeilberger, DK, Hofer, Fulman, Feray–Kammoun) but also provides a combinatorial, representation‑theory‑free pathway to understanding local permutation statistics across cycle types with wide applicability.

Abstract

Beginning with work of Zeilberger on classical pattern counts, there are a variety of structural results for moments of permutation statistics applied to random permutations. Using tools from representation theory, Gaetz and Ryba generalized Zeilberger's results to uniformly random permutations of a given cycle type. We introduce regular statistics and characterize their moments for all cycle types, generalizing all results in this literature that we are aware of. Our approach splits into two steps: first characterize such statistics as linear combinations of indicator functions for partial permutations, then identifying the moments of such indicators. As an application, we show that many regular statistics exhibit a law of large numbers depending only on fixed point counts and a similar variance property that depends also on two--cycle counts. These results first appeared in arXiv:2206.06567, which is no longer intended for publication. Our original proof of the moment result for indicators of partial permutations relied on representation theory and symmetric functions. A referee generously shared a combinatorial argument, allowing us to give a self-contained treatment of these results that does not rely on representation theory.

Moments of permutation statistics by cycle type

TL;DR

The work develops a unified framework to analyze moments and asymptotics of permutation statistics conditioned on cycle type. By expressing statistics as linear combinations of partial permutation indicators and exploiting Möbius inversion, the authors prove a key polynomiality result: for a partial permutation of size with cycle‑path type , equals a polynomial . They introduce regular statistics, show their moments are polynomials with explicit degree bounds, and demonstrate closure under multiplication, enabling a broad moment theory that includes classical and bivincular pattern counts. The paper derives asymptotic behavior where the mean and variance depend only on fixed points and two‑cycles, establishes LLNs and permuton limits, and discusses extensions to other groups and combinatorial objects. This framework not only generalizes prior results (Zeilberger, DK, Hofer, Fulman, Feray–Kammoun) but also provides a combinatorial, representation‑theory‑free pathway to understanding local permutation statistics across cycle types with wide applicability.

Abstract

Beginning with work of Zeilberger on classical pattern counts, there are a variety of structural results for moments of permutation statistics applied to random permutations. Using tools from representation theory, Gaetz and Ryba generalized Zeilberger's results to uniformly random permutations of a given cycle type. We introduce regular statistics and characterize their moments for all cycle types, generalizing all results in this literature that we are aware of. Our approach splits into two steps: first characterize such statistics as linear combinations of indicator functions for partial permutations, then identifying the moments of such indicators. As an application, we show that many regular statistics exhibit a law of large numbers depending only on fixed point counts and a similar variance property that depends also on two--cycle counts. These results first appeared in arXiv:2206.06567, which is no longer intended for publication. Our original proof of the moment result for indicators of partial permutations relied on representation theory and symmetric functions. A referee generously shared a combinatorial argument, allowing us to give a self-contained treatment of these results that does not rely on representation theory.

Paper Structure

This paper contains 17 sections, 23 theorems, 90 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Partial Permutations
  4. Probability and Permutation Statistics
  5. Expectation for Partial Permutations by Cycle Type
  6. Regular Statistics
  7. Proof of Theorem \ref{['t:regular-moment']}
  8. Examples of Regular Statistics
  9. Asymptotic Results
  10. Asymptotics of Regular Statistics
  11. Convergence for long cycles
  12. Permutons
  13. Final Remarks
  14. Other groups
  15. Joint distributions
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $(I,J)$ be a partial permutation of size $k$ with cycle--path type $(\mu,\nu)$ and $|I \cup J| = m$. Then there is a polynomial $f_{(\mu,\nu)} \in {\mathbb {Z}}[n,m_1,\dots,m_k]$ of degree $k$ where $\deg n = 1$ and $\deg m_i = i$ so that for any partition $\lambda$

Figures (1)

  • Figure 1: For $I = (2,3,5,7,9,10,11,13,15), J = (7,9,5,14,3,6,10,1,15) \in {\mathfrak{S}}_{15,9}$ we depict the graph $G(I,J)$. The cycle and path partition are $(2,1,1)$ and $(2,2,1)$.

Theorems & Definitions (49)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 2.1
  • Proposition 2.2
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • ...and 39 more