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Limit Theorems for Fixed Point Biased Pattern Avoiding Involutions

Jungeun Park, Douglas Rizzolo

TL;DR

This work analyzes the distribution of fixed points in fixed-point biased involutions that avoid a pattern, extending prior results for length-3 patterns to the involution setting. It develops a unified analytic-combinatorics framework based on bivariate generating functions $G(z,q)$ and a density relationship between the biased and unbiased measures to derive limiting laws across all patterns of length $3$, including phase transitions in the bias parameter $q$. For monotone and certain long patterns, the paper provides explicit limiting distributions, such as parity-conditioned Negative Binomial limits, GOE-tilted limits, Rayleigh-type boundaries, and Gaussian limits with explicit centering and variance, depending on the pattern and regime of $q$. These results deepen understanding of how pattern avoidance couples with fixed-point bias to shape the asymptotic fixed-point structure, with connections to random matrix theory via GOE-type limits and to classical limit theorems through analytic combinatorics.

Abstract

We study fixed point biased involutions that avoid a pattern. For every pattern of length three we obtain limit theorems for the asymptotic distribution of the (appropriately centered and scaled) number of fixed points of a random fixed point biased involution avoiding that pattern. When the pattern being avoided is either $321$, $132$, or $213$, we find a phase transition depending on the strength of the bias. We also obtain a limit theorem for distribution of fixed points when the pattern is $123\cdots k(k+1)$ for any $k$ and partial results when the pattern is $(k+1)k\cdots 321$.

Limit Theorems for Fixed Point Biased Pattern Avoiding Involutions

TL;DR

This work analyzes the distribution of fixed points in fixed-point biased involutions that avoid a pattern, extending prior results for length-3 patterns to the involution setting. It develops a unified analytic-combinatorics framework based on bivariate generating functions and a density relationship between the biased and unbiased measures to derive limiting laws across all patterns of length , including phase transitions in the bias parameter . For monotone and certain long patterns, the paper provides explicit limiting distributions, such as parity-conditioned Negative Binomial limits, GOE-tilted limits, Rayleigh-type boundaries, and Gaussian limits with explicit centering and variance, depending on the pattern and regime of . These results deepen understanding of how pattern avoidance couples with fixed-point bias to shape the asymptotic fixed-point structure, with connections to random matrix theory via GOE-type limits and to classical limit theorems through analytic combinatorics.

Abstract

We study fixed point biased involutions that avoid a pattern. For every pattern of length three we obtain limit theorems for the asymptotic distribution of the (appropriately centered and scaled) number of fixed points of a random fixed point biased involution avoiding that pattern. When the pattern being avoided is either , , or , we find a phase transition depending on the strength of the bias. We also obtain a limit theorem for distribution of fixed points when the pattern is for any and partial results when the pattern is .
Paper Structure (3 sections, 4 theorems, 37 equations)

This paper contains 3 sections, 4 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Proofs

Key Result

Theorem 1

Fix $k\geq 1$, $q>0$, $\sigma =123\cdots k(k+1)$. Suppose that $\Pi_n$ is a random element of $\textbf{I\!v}_n(\sigma)$ distributed according to $\mathbb{P}^{q,\sigma}_{n}$. Then and where $X^q_{even}$ has probability mass function and

Theorems & Definitions (7)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • proof : Proof of Theorem \ref{['thm inc']}
  • proof : Proof of Theorem \ref{['thm dec']}
  • proof : Proof of Theorem \ref{['thm phase']}