Eulerian Numbers Associated with Arithmetical Progressions

In this paper, we give a combinatorial interpretation of the r-Whitney-Eulerian numbers by means of coloured signed permutations. This sequence is a generalization of the well-known Eulerian numbers and it is connected to r-Whitney numbers of the second kind. Using generating functions, we provide some combinatorial identities and the log-concavity property. Finally, we show some basic congruences involving the r-Whitney-Eulerian numbers.