A combinatorial approach to the generalized central factorial numbers
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.
scientific article; zbMATH DE number 7388045
scientific article