Second Best, Third Worst, Fourth in Line

TL;DR

This paper explores the properties of decomposable combinatorial structures, analyzing the distribution of cycle lengths and components, and compares theoretical predictions with experimental data to identify gaps and open questions.

Contribution

It provides new insights into the behavior of the exp-log class of combinatorial structures, especially regarding the second longest cycle and second smallest component.

Findings

The modal length of the second longest cycle in a random permutation is approximately 0.2350 times n.

The modal length of the second smallest component in a random mapping is 2.

The study highlights discrepancies between theory and experimental data, raising open questions.

Abstract

We investigate decomposable combinatorial labeled structures more fully, focusing on the exp-log class of type a=1 or 1/2. For instance, the modal length of the second longest cycle in a random n-permutation is (0.2350...)n, whereas the modal length of the second smallest component in a random n-mapping is 2 (conjecturally, given n>=434). As in earlier work, our approach is to establish how well existing theory matches experimental data and to raise open questions.