A note on the Screaming Toes game

TL;DR

This paper explores the properties of random mappings with deranged cores, analyzing how they differ from classical permutations, especially in small component behavior, with explicit results illustrating these differences.

Contribution

It introduces a new framework for studying the Screaming Toes game using deranged cores in random mappings, highlighting key differences from traditional permutation models.

Findings

Distinct behavior of small components in deranged core mappings

Explicit results illustrating differences from classical permutation models

Insights into the structure of mappings relevant to the Screaming Toes game

Abstract

We investigate properties of random mappings whose core is composed of derangements as opposed to permutations. Such mappings arise as the natural framework to study the Screaming Toes game described, for example, by Peter Cameron. This mapping differs from the classical case primarily in the behaviour of the small components, and a number of explicit results are provided to illustrate these differences.