Limit theorems for fixed point biased permutations avoiding a pattern of length three

TL;DR

The paper analyzes fixed points in permutations under two generalizations: a Gibbs-type fixed-point bias with parameter $q$ and the constraint of pattern avoidance. It shows that, without pattern avoidance, the fixed-point count converges to Poisson$(q)$, while with pattern avoidance the limiting distribution undergoes a phase transition as $q$ crosses 3. For patterns $\{132,321,213\}$, the limits are NegativeBinomial$(2,1-\frac{q}{3})$ for $q<3$, Rayleigh$(\tfrac{3}{\sqrt{2}})$ after $\sqrt{n}$-scaling at $q=3$, and Normal after centering and scaling for $q>3$. The asymptotics of the normalization constants $Z_n(q,\tau)$ are obtained via singularity analysis of a bivariate generating function, revealing the mechanism behind the phase transition. The work connects Gibbs-biased permutation models with pattern-avoiding combinatorics, and outlines future directions toward other patterns, joint statistics, and permuton descriptions.

Abstract

We prove limit theorems for the number of fixed points occurring in a random pattern-avoiding permutation distributed according to a one-parameter family of biased distributions. The bias parameter exponentially tilts the distribution towards favoring permutations with more or fewer fixed points than is typical under the uniform distribution. One case we study features a phase transition where the limiting distribution changes abruptly from negative binomial to Rayleigh to normal depending on the bias parameter.

Theorems & Definitions (13)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Lemma 1
• proof : Proof of Lemma \ref{['lem:normalization']}
• proof : Proof of Theorem \ref{['thm:classic']}
• proof : Proof of Theorem \ref{['thm:Bernoulli']}
• proof : Proof of Theorem \ref{['thm:subcritical']}