What question did this study set out to answer?

This research aims to understand the number of fixed points in biased pattern-avoiding permutations.

March 12, 2026Open Access

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

Key Points

This research aims to understand the number of fixed points in biased pattern-avoiding permutations.
Analyzed a one-parameter family of biased distributions
Examined fixed point occurrences in random permutations
Studied the impact of bias parameters on distribution types
Identified a phase transition in limiting distributions
Observed transitions from negative binomial to Rayleigh to normal depending on bias
Demonstrated how bias influences the fixed points in permutations

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. 17 pages, final version

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