Associated $r$-Dowling numbers and some relatives

Gyimesi, Eszter; Nyul, Gábor

Comptes Rendus. Mathématique, Tome 359 (2021) no. 1, pp. 47-55.

In this paper, we introduce a new generalization of Bell numbers, the $s$-associated $r$-Dowling numbers by combining two investigational directions. Here, $r$ distinguished elements have to be in distinct blocks, some elements are coloured according to a colouring rule, and the cardinality of certain blocks is bounded from below by $s$. Along with them, we define some relatives, the $s$-associated $r$-Dowling factorials and the $s$-associated $r$-Dowling–Lah numbers, when the underlying set is decomposed into cycles or ordered blocks. The study of these numbers is based on their combinatorial meaning, and the exponential generating functions of their sequences derived from the so-called $r$-compositional formula.

Classification : 05A15, 05A18, 05A19, 11B73

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