A combinatorial approach to the generalized central factorial numbers
type: Item
instance of: scholarly article
title: A combinatorial approach to the generalized central factorial numbers
zbMATH Open document ID: 1486.11028
author: Takao Komatsu
author: José L. Ramírez
author: Diego Villamizar
published in: Mediterranean Journal of Mathematics
publication date: 2021-08-30T00:00:00Z
review text: The combinatorial interpretation for the Genocchi numbers given by \textit{D. Dumont} [Duke Math. J. 41, 305--318 (1974; Zbl 0297.05004)], \textit{M. Domaratzki} [J. Integer Seq. 7, No. 3, Art. 04.3.6, 11 p. (2004; Zbl 1092.11010)] shows that the sequence $T_s (n,k)$, the generalized factorial numbers, counts the number of $s$-tuples of quasi-permutations of $[n]:=\{1, 2, \ldots, n\}$ under certain conditions. In the present paper, the authors provide a simple combinatorial interpretation for the generalized central factorial numbers $T_s (n, k)$ using set partitions. By means of this combinatorial interpretation, some combinatorial properties are investigated. A relation with the Stirling numbers of the first kind with higher level is derived. The authors show that these two sequences preserve the orthogonality relation between the classical Stirling numbers of the first kind and the second kind. Finally, a connection between the generalized central factorial numbers and the Bernoulli numbers is provided.
Mathematics Subject Classification ID: 11B65
Mathematics Subject Classification ID: 11B68
Mathematics Subject Classification ID: 11B73
Mathematics Subject Classification ID: 05A15
zbMATH DE Number: 7388045
zbMATH Keywords: central factorial numbers
zbMATH Keywords: set partitions
zbMATH Keywords: Bernoulli numbers
zbMATH Keywords: generating functions
zbMATH Keywords: Stirling numbers
reviewed by: Ugur Duran
MaRDI profile type: Publication
full work available at URL: s00009-021-01830-5
OpenAlex ID: W3195496662
cites work: The \(r\)-Stirling numbers
cites work: Central factorial numbers; their main properties and some applications.
cites work: The Divided Central Differences of Zero
cites work: Q3024007
cites work: Interpretations combinatoires des nombres de Genocchi
cites work: A Combinatorial Approach to the Stirling Numbers of the First Kind with Higher Level
cites work: Combinatorics of Set Partitions
cites work: Connections between central factorial numbers and Bernoulli polynomials
cites work: Combinatorics and Number Theory of Counting Sequences
cites work: Q5589310
DOI: 10.1007/S00009-021-01830-5
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description: scientific article; zbMATH DE number 7388045