A181731 Table A(d,n) of the number of paths of a chess rook in a d-dimensional hypercube from (0...0) to (n...n) where the rook may move in steps that are multiples of (1,0..0), (0,1,0..0), ..., (0..0,1).
1, 1, 1, 1, 2, 2, 1, 6, 14, 4, 1, 24, 222, 106, 8, 1, 120, 6384, 9918, 838, 16, 1, 720, 291720, 2306904, 486924, 6802, 32, 1, 5040, 19445040, 1085674320, 964948464, 25267236, 56190, 64, 1, 40320, 1781750880, 906140159280, 4927561419120, 439331916888, 1359631776, 470010, 128, 1, 362880, 214899027840, 1224777388630320, 54259623434853360
Offset: 1
Examples
A(3,1) = 6 because there are 6 rook paths on 3D chessboards from (0,0,0) to (1,1,1). Square table A(d,n) begins: 1, 1, 2, 4, 8, ... 1, 2, 14, 106, 838, ... 1, 6, 222, 9918, 486924, ... 1, 24, 6384, 2306904, 964948464, ... 1, 120, 291720, 1085674320, 4927561419120, ...
Links
- Alois P. Heinz, Antidiagonals n = 1..20
- M. Kauers and D. Zeilberger, The Computational Challenge of Enumerating High-Dimensional Rook Walks, arXiv:1011.4671 [math.CO], 2010.
Crossrefs
Programs
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Maple
b:= proc(l) option remember; `if`({l[]} minus {0}={}, 1, add(add (b(sort(subsop(i=l[i]-j, l))), j=1..l[i]), i=1..nops(l))) end: A:= (d, n)-> b([n$d]): seq(seq(A(h-n, n), n=0..h-1), h=1..10); # Alois P. Heinz, Jul 21 2012 -
Mathematica
b[l_List] := b[l] = If[Union[l] ~Complement~ {0} == {}, 1, Sum[ Sum[ b[ Sort[ ReplacePart[l, i -> l[[i]] - j]]], {j, 1, l[[i]]}], {i, 1, Length[l]}]]; A[d_, n_] := b[Array[n&, d]]; Table[Table[A[h-n, n], {n, 0, h-1}], {h, 1, 10}] // Flatten (* Jean-François Alcover, Feb 25 2015, after Alois P. Heinz *)
Extensions
Edited by Alois P. Heinz, Jul 21 2012
Minor edits by Vaclav Kotesovec, Sep 03 2014
Comments