Fixed points of a random restricted growth sequence

TL;DR

This paper investigates fixed points in restricted growth sequences, using generating functions and probabilistic sampling to enumerate and analyze their properties.

Contribution

It introduces a novel approach combining generating functions and probabilistic sampling to count fixed points in restricted growth sequences.

Findings

Derived explicit formulas for fixed point enumeration

Developed a probabilistic sampling method for analysis

Provided insights into the distribution of fixed points

Abstract

We call $i$ a fixed point of a given sequence if the value of that sequence at the $i$-th position coincides with $i$. Here, we enumerate fixed points in the class of restricted growth sequences. The counting process is conducted by calculation of generating functions and leveraging a probabilistic sampling method.