We consider a generalization of Eulerian numbers which count the number of placements of rooks on an chessboard where there are exactly rooks in each row and each column, and exactly rooks below the main diagonal. The standard Eulerian numbers correspond to the case . We show that for any the resulting numbers are symmetric and give generating functions of these numbers for small values of .
"A generalization of Eulerian numbers via rook placements." Involve 10 (4) 691 - 705, 2017. https://doi.org/10.2140/involve.2017.10.691