A generalization of Eulerian numbers via rook placements

@article{Banaian2015AGO,
  title={A generalization of Eulerian numbers via rook placements},
  author={Esther Banaian and Steve Butler and Christopher Cox and Jeffrey Davis and Jacob Landgraf and Scarlitte Ponce},
  journal={arXiv: Combinatorics},
  year={2015}
}
We consider a generalization of Eulerian numbers which count the number of placements of $cn$ "rooks" on an $n\times n$ board where there are exactly $c$ rooks in each row and each column, and exactly $k$ rooks below the main diagonal. The standard Eulerian numbers correspond to the case $c=1$. We show that for any $c$ the resulting numbers are symmetric and give generating functions of these numbers for small values of $k$. 

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