Limit Theorems for Fixed Point Biased Pattern Avoiding Involutions

TL;DR

This paper investigates the asymptotic behavior of fixed points in pattern-avoiding involutions with bias, revealing phase transitions for certain patterns and establishing limit theorems for others.

Contribution

It provides new limit theorems for fixed points in biased involutions avoiding specific patterns, including phase transition phenomena for patterns of length three.

Findings

Limit theorems for fixed points in pattern-avoiding involutions.

Phase transition depending on bias strength for patterns 321, 132, 213.

Asymptotic distribution results for patterns 123...k and partial results for (k+1)k...321.

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 $(k+1)k\cdots 321$.