On the Number of Conjugate Classes of Derangements

Wen-Wei Li,Zhong-Lin Cheng,Jia-Bao Liu,Gaetano Luciano

doi:10.1155/2021/6023081

Wen-Wei Li, Zhong-Lin Cheng + Show 2 more

Open Access

https://doi.org/10.1155/2021/6023081

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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 × 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 and 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 a 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.

Highlights

N is a positive integer greater than 1
When generating the representatives of all the isotopy classes of Latin rectangles of order n by some method, we need to know the number of the isotopy classes of 2 × n Latin rectangles for verification
We need the approximate value in a simple and efficient method. (When writing a C program to generate the representatives of all the isotopy classes of Latin rectangles of order n, we need to prepare some space in the memory module (RAM) to store the cycle structures of derangements so as to make the program more efficient; otherwise, we have to allocate memory dynamically, which will cost more time in memory addressing when writing and reading data frequently in the particular position in the memory module

Summary

Introduction

N is a positive integer greater than 1. A cycle structure of a derangement of order n could be considered as an integer solution of the equation as follows: s1 + s2 + · · · + sq n, 􏼐2 ≤ s1 ≤ s2 ≤ · · · ≤ sq􏼑, (2). Q 1 ere is a brief introduction of the important results on the partition number (or partition function) p(n) and Pq(n) in reference [2], such as the recursion formula of p(n) and Pq(n). Sometimes we need the approximation value, such as the cases mentioned in [1], so an estimation formula is necessary. E author has not found a practical estimation formula with high accuracy of the number h(n) before. Half a year after the main results were obtained in this paper, the author found an asymptotic formula as follows:. It is necessary to modify the asymptotic formula for better accuracy

Method A

Method C

Findings

Conclusion