Eulerian Numbers Associated with Arithmetical Progressions

José L. Ramírez, Sergio N. Villamarin, Diego Villamizar


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.


Eulerian number; $r$-Whitney number; $r$-Whitney-Eulerian number; Combinatorial identities; Unimodality

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