Permute, Graph, Map, Derange

TL;DR

This paper investigates the properties of certain combinatorial structures, comparing theoretical predictions with experimental data, and highlights open problems in the study of permutations and mappings.

Contribution

It provides an empirical analysis of decomposable combinatorial structures in the exp-log class, comparing theory with experimental results for permutations and mappings.

Findings

Median longest cycle in permutation is approximately 60.65% of n

Median largest component in mapping is approximately 78.64% of n

Highlights open problems in combinatorial structure analysis

Abstract

We study decomposable combinatorial labeled structures in the exp-log class, specifically, two examples of type a=1 and two examples of type a=1/2. Our approach is to establish how well existing theory matches experimental data. For instance, the median length of the longest cycle in a random n-permutation is (0.6065...)*n, whereas the median length of the largest component in a random n-mapping is (0.7864...)*n. Unsolved problems are highlighted, in the hope that someone else might address these someday.