Associated r-Dowling numbers and some relatives

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. 2020 Mathematics Subject Classification. 05A15, 05A18, 05A19, 11B73. Funding. Research of the first author Eszter Gyimesi was supported by the ÚNKP-18-3 New National Excellence Program of the Ministry of Human Capacities. Research of the second author Gábor Nyul was supported by Grant 115479 from the Hungarian Scientific Research Fund. Manuscript received 22nd July 2019, revised 17th February 2020, accepted 5th November 2020.