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A note on some identities of derangement polynomials. (English) Zbl 1383.05025

Summary: The problem of counting derangements was initiated by Pierre Rémond de Montmort in 1708 (see [L. Carlitz, Fibonacci Q. 16, 255–258 (1978; Zbl 0403.05008); R. J. Clarke and M. Sved, Math. Mag. 66, No. 5, 299–303 (1993; Zbl 0837.05013); T. Kim et al., “Fourier series of \(r\)-derangement and higher-order derangement functions”, Adv. Stud. Contemp. Math., Kyungshang 28, No. 1, 1–11 (2018; doi:10.17777/ascm2018.28.1.1)]). A derangement is a permutation that has no fixed points, and the derangement number \(d_{n}\) 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.

MSC:

05A19 Combinatorial identities, bijective combinatorics
05A40 Umbral calculus
11B73 Bell and Stirling numbers
11B83 Special sequences and polynomials
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References:

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