On the Number of Conjugate Classes of Derangements

TL;DR

This paper introduces a recursive and highly accurate approximation formula for the number of conjugate classes of derangements, which is useful for practical calculations in engineering and experimental data analysis.

Contribution

It provides a new recursion formula and elementary approximation methods for calculating the conjugate classes of derangements efficiently.

Findings

Derived a recursion formula for h(n).

Presented high-accuracy approximation formulas.

Facilitated practical calculations without programming.

Abstract

The number of conjugate classes of derangements of order $n$ is the same as the number $h(n)$ of the restricted partitions with every portion greater than $1$. It is also equal to the number of isotopy classes of $2\times n$ Latin rectangles. Sometimes the exact value is necessary, while sometimes we need the approximation value. In this paper, a recursion formula of $h(n)$ will be obtained, also will some elementary approximation formulae with high accuracy for $h(n)$ be presented. Although we may obtain the value of $h(n)$ in some computer algebra system, it is still meaningful to find an efficient way to calculate the approximate value, especially in engineering, since most people are familiar with neither programming nor CAS software. This paper is mainly for the readers who need a simple and practical formula to obtain the approximate value (without writing a program) with more accuracy, such as to compute the value in an pocket science calculator without programming function. Some methods used here can also be applied to find the fitting functions for some types of data obtained in experiments.