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A note on some identities of derangement polynomials

The problem of counting derangements was initiated by Pierre Rémond de Montmort in 1708 (see Carlitz in Fibonacci Q. 16(3):255–258, 1978, Clarke and Sved in Math. Mag. 66(5):299–303, 1993, Kim, Kim and Kwon in Adv. Stud. Contemp. Math. (Kyungshang) 28(1):1–11 2018. A derangement is a permutation tha... Full description

Journal Title: Journal of Inequalities and Applications 2018, Vol.2018(1), pp.1-17
Main Author: Kim, Taekyun
Other Authors: Kim, Dae , Jang, Gwan-Woo , Kwon, Jongkyum
Format: Electronic Article Electronic Article
Language: English
Subjects:
ID: E-ISSN: 1029-242X ; DOI: 10.1186/s13660-018-1636-8
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recordid: springer_jour10.1186/s13660-018-1636-8
title: A note on some identities of derangement polynomials
format: Article
creator:
  • Kim, Taekyun
  • Kim, Dae
  • Jang, Gwan-Woo
  • Kwon, Jongkyum
subjects:
  • Derangement numbers
  • Derangement polynomials
  • -derangement numbers
  • -derangement polynomials
  • Umbral calculus
ispartof: Journal of Inequalities and Applications, 2018, Vol.2018(1), pp.1-17
description: The problem of counting derangements was initiated by Pierre Rémond de Montmort in 1708 (see Carlitz in Fibonacci Q. 16(3):255–258, 1978, Clarke and Sved in Math. Mag. 66(5):299–303, 1993, Kim, Kim and Kwon in Adv. Stud. Contemp. Math. (Kyungshang) 28(1):1–11 2018. A derangement is a permutation that has no fixed points, and the derangement number d n $d_{n}$ is the number of fixed-point-free permutations on an n element set. In this paper, we study the derangement polynomials and investigate some interesting properties which are related to derangement numbers. Also, we study two generalizations of derangement polynomials, namely higher-order and r -derangement polynomials, and show some relations between them. In addition, we express several special polynomials in terms of the higher-order derangement polynomials by using umbral calculus.
language: eng
source:
identifier: E-ISSN: 1029-242X ; DOI: 10.1186/s13660-018-1636-8
fulltext: fulltext_linktorsrc
issn:
  • 1029-242X
  • 1029242X
url: Link


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subjectDerangement numbers ; Derangement polynomials ; -derangement numbers ; -derangement polynomials ; Umbral calculus
descriptionThe problem of counting derangements was initiated by Pierre Rémond de Montmort in 1708 (see Carlitz in Fibonacci Q. 16(3):255–258, 1978, Clarke and Sved in Math. Mag. 66(5):299–303, 1993, Kim, Kim and Kwon in Adv. Stud. Contemp. Math. (Kyungshang) 28(1):1–11 2018. A derangement is a permutation that has no fixed points, and the derangement number d n $d_{n}$ is the number of fixed-point-free permutations on an n element set. In this paper, we study the derangement polynomials and investigate some interesting properties which are related to derangement numbers. Also, we study two generalizations of derangement polynomials, namely higher-order and r -derangement polynomials, and show some relations between them. In addition, we express several special polynomials in terms of the higher-order derangement polynomials by using umbral calculus.
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titleA note on some identities of derangement polynomials
descriptionThe problem of counting derangements was initiated by Pierre Rémond de Montmort in 1708 (see Carlitz in Fibonacci Q. 16(3):255–258, 1978, Clarke and Sved in Math. Mag. 66(5):299–303, 1993, Kim, Kim and Kwon in Adv. Stud. Contemp. Math. (Kyungshang) 28(1):1–11 2018. A derangement is a permutation that has no fixed points, and the derangement number d n $d_{n}$ is the number of fixed-point-free permutations on an n element set. In this paper, we study the derangement polynomials and investigate some interesting properties which are related to derangement numbers. Also, we study two generalizations of derangement polynomials, namely higher-order and r -derangement polynomials, and show some relations between them. In addition, we express several special polynomials in terms of the higher-order derangement polynomials by using umbral calculus.
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abstractThe problem of counting derangements was initiated by Pierre Rémond de Montmort in 1708 (see Carlitz in Fibonacci Q. 16(3):255–258, 1978, Clarke and Sved in Math. Mag. 66(5):299–303, 1993, Kim, Kim and Kwon in Adv. Stud. Contemp. Math. (Kyungshang) 28(1):1–11 2018. A derangement is a permutation that has no fixed points, and the derangement number d n $d_{n}$ is the number of fixed-point-free permutations on an n element set. In this paper, we study the derangement polynomials and investigate some interesting properties which are related to derangement numbers. Also, we study two generalizations of derangement polynomials, namely higher-order and r -derangement polynomials, and show some relations between them. In addition, we express several special polynomials in terms of the higher-order derangement polynomials by using umbral calculus.
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