# Combinatorial study of $r$-Whitney and $r$-Dowling numbers

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Eszter Gyimesi

Institute of Mathematics, University of Debrecen

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G\'{a}bor Nyul

Institute of Mathematics, University of Debrecen

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**Minisymposium:**
COMBINATORICS

**Content:**
T. A. Dowling introduced Whitney numbers of the first and second kind concerning the so-called Dowling lattices of finite groups. It turned out that they are generalizations of Stirling numbers. Later, I. Mez\H{o} defined $r$-Whitney numbers as a common generalization of Whitney numbers and $r$-Stirling numbers.
By summation of $r$-Whitney numbers of the second kind, we obtain $r$-Dowling numbers which are therefore the generalizations of Bell numbers.
In our talk, we present a new combinatorial interpretation of $r$-Whitney and $r$-Dowling numbers. This allows us to explain their properties in a purely combinatorial manner, as well as derive some new identities.