A generalization of Eulerian numbers via rook placements

Banaian, Esther; Butler, Steve; Cox, Christopher; Davis, Jeffrey; Landgraf, Jacob; Ponce, Scarlitte

doi:10.2140/involve.2017.10.691

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A generalization of Eulerian numbers via rook placements

Esther Banaian, Steve Butler, Christopher Cox, Jeffrey Davis, Jacob Landgraf and Scarlitte Ponce

Vol. 10 (2017), No. 4, 691–705

DOI: 10.2140/involve.2017.10.691

Abstract

We consider a generalization of Eulerian numbers which count the number of placements of $c n$ rooks on an $n \times n$ chessboard 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 $k$ .

Keywords

Eulerian numbers, juggling, recursion, multiplex

Mathematical Subject Classification 2010

Primary: 05A15

Milestones

Received: 26 April 2016

Accepted: 11 July 2016

Published: 7 March 2017

Communicated by Jim Haglund

Authors

Esther Banaian
College of St. Benedict Collegeville, MN 56321 United States

Steve Butler
Department of Mathematics Iowa State University Ames, IA 50011 United States

Christopher Cox
Department of Mathematical Sciences Carnegie Mellon University Pittsburgh, PA 15213 United States

Jeffrey Davis
Department of Mathematics and Computer Science Emory University Atlanta, GA 30322 United States

Jacob Landgraf
Department of Mathematics University of Notre Dame Notre Dame, IN 46556 United States

Scarlitte Ponce
Department of Mathematics Iowa State University Ames, IA 50011 United States

Vol. 10, No. 4, 2017