Vol. 6, No. 1, 2013

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Rook polynomials in three and higher dimensions

Feryal Alayont and Nicholas Krzywonos

Vol. 6 (2013), No. 1, 35–52
Abstract

The rook polynomial of a board counts the number of ways of placing nonattacking rooks on the board. In this paper, we describe how the properties of the two-dimensional rook polynomials generalize to the rook polynomials of “boards” in three and higher dimensions. We also define families of three-dimensional boards which generalize the two-dimensional triangle boards and the boards representing the problème des rencontres. The rook coefficients of these three-dimensional boards are shown to be related to famous number sequences such as the central factorial numbers, the number of Latin rectangles and the Genocchi numbers.

Keywords
rook polynomial, three dimensions, central factorial numbers, Genocchi numbers, problème des rencontres
Mathematical Subject Classification 2010
Primary: 05A05, 05A10
Secondary: 11B73
Milestones
Received: 15 September 2011
Accepted: 30 May 2012
Published: 23 June 2013

Communicated by Jim Haglund
Authors
Feryal Alayont
Department of Mathematics
Grand Valley State University
1 Campus Drive
Allendale, MI 49401
United States
Nicholas Krzywonos
Department of Mathematics
Grand Valley State University
1 Campus Drive
Allendale, MI 49401
United States