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A note on some identities of derangement polynomials
Taekyun Kim1,2, Dae San Kim3, Gwan-Woo Jang2
11Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin, China.
Journal of Inequalities and Applications
|March 3, 2018
Summary
This study explores derangement polynomials, which count permutations with no fixed points. Researchers investigated properties of these polynomials and their generalizations, including higher-order and r-derangement polynomials.
Area of Science:
- Combinatorics
- Discrete Mathematics
- Number Theory
Background:
- The concept of derangements, permutations without fixed points, was introduced by Pierre Rémond de Montmort in 1708.
- Derangement numbers, denoted as [Formula: see text], quantify fixed-point-free permutations on an n-element set.
Purpose of the Study:
- To investigate the properties of derangement polynomials.
- To study two generalizations: higher-order derangement polynomials and r-derangement polynomials.
- To establish relationships between these generalized polynomials and derangement numbers.
Main Methods:
- Analysis of derangement polynomials and their properties.
- Exploration of higher-order and r-derangement polynomials.
- Application of umbral calculus to express special polynomials in terms of higher-order derangement polynomials.
Main Results:
- Identification of interesting properties related to derangement numbers.
- Demonstration of relations between higher-order and r-derangement polynomials.
- Expression of special polynomials using higher-order derangement polynomials via umbral calculus.
Conclusions:
- The study provides new insights into the combinatorial properties of derangement polynomials and their generalizations.
- The findings contribute to the understanding of permutations and fixed-point-free arrangements.
- Umbral calculus offers a powerful tool for relating different polynomial families in combinatorics.
Keywords:
Derangement numbersDerangement polynomialsUmbral calculusr-derangement numbersr-derangement polynomialsMore Related Videos
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