Rook Polynomial

Author

Eric W. Weisstein
May 30, 2006

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Combinatorics/RookPolynomial.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/RookPolynomial.html.

©2006 Wolfram Research, Inc. except for portions noted otherwise

Initialization

<<DiscreteMath`Combinatorica`

Notes

The routine ChessPieces[] in Chess.m needs to be extended to allow a 2x2 board.

Rectangular Boards

RookPolynomial[n_Integer,m_Integer,x_]:=Expand[n!x^n LaguerreL[n,m-n,-1/x]]

RookPolynomial[2,5,x]

2x2 Illustration

Show[GraphicsArray[{
{MagicSquarePlot[{{"",""},{"",""}},DisplayFunction->Identity]},
MagicSquarePlot[Partition[#,2],DisplayFunction->Identity]&/@Permutations[{"\!\(\*
StyleBox[\"r\",\nFontFamily->\"ChessLeipzig\",\nFontSize->36]\)","","",""}],
{
MagicSquarePlot[{{"\!\(\*
StyleBox[\"r\",\nFontFamily->\"ChessLeipzig\",\nFontSize->36]\)",""},{"","\!\(\*
StyleBox[\"r\",\nFontFamily->\"ChessLeipzig\",\nFontSize->36]\)"}},DisplayFunction->Identity],
MagicSquarePlot[{{"","\!\(\*
StyleBox[\"r\",\nFontFamily->\"ChessLeipzig\",\nFontSize->36]\)"},{"\!\(\*
StyleBox[\"r\",\nFontFamily->\"ChessLeipzig\",\nFontSize->36]\)",""}},DisplayFunction->Identity]
}
}]]

Square Boards

Table[Expand[n!LaguerreL[n,x]],{n,4}]//ColumnForm

Table[Expand[n!x^n LaguerreL[n,-1/x]],{n,4}]//ColumnForm

Table[RookPolynomial[n,n,x],{n,8}]//ColumnForm//TraditionalForm

Flatten[Table[CoefficientList[RookPolynomial[n,n,x],x],{n,8}]]

ID Number: A021010
Sequence:  1,1,1,1,4,2,1,9,18,6,1,16,72,96,24,1,25,200,600,600,120,1,
           36,450,2400,5400,4320,720,1,49,882,7350,29400,52920,35280,
           5040,1,64,1568,18816,117600,376320,564480,322560,40320,1,81,
           2592
Signed:    1,-1,1,1,-4,2,-1,9,-18,6,1,-16,72,-96,24,-1,25,-200,600,
           -600,120,1,-36,450,-2400,5400,-4320,720,-1,49,-882,7350,
           -29400,52920,-35280,5040,1,-64,1568,-18816,117600,-376320,
           564480,-322560,40320,-1,81,-2592
Name:      Triangle of coefficients of Laguerre polynomials L_n(x).
References M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical
              Functions, National Bureau of Standards Applied Math. Series 55, 1964
              (and various reprintings), p. 799.
Links:     Index entries for sequences related to Laguerre polynomials
Example:   1; -1,1; 1,-4,2; -1,9,-18,6; 1,-16,72,-96,24; ...
See also:  Cf. A021009.
Keywords:  sign,done,tabl,easy,nice
Offset:    0
Author(s): njas
Extension: More terms from James A. Sellers (sellersj@math.psu.edu)

Table[Binomial[k,k](-1)^k Pochhammer[-k,k],{k,8}]

Table[Binomial[k,k]^2k!,{k,8}]

Table[Coefficient[RookPolynomial[n,n,x],x,n],{n,8}]

Table[n!,{n,8}]