A087074 For n > 0, 0 <= k <= n^2, T(n,k) is the number of rotationally and reflectively distinct n X n arrays that contain the numbers 1 through k once each and n^2-k zeros.

1, 1, 1, 1, 2, 3, 3, 1, 3, 12, 66, 378, 1890, 7560, 22680, 45360, 45360, 1, 3, 33, 426, 5466, 65520, 720720, 7207200, 64864800, 518918400, 3632428800, 21794572800, 108972864000, 435891456000, 1307674368000, 2615348736000, 2615348736000, 1

Offset: 1

Author

Zak Seidov and Eric W. Weisstein, Aug 08 2003

Keywords

• nonn
• tabf

Examples

			1,1; 1,1,2,3,3; 1,3,12,66,378,1890,7560,22680,45360,45360; ...
There is a single distinct 3 X 3 matrix containing all zeros, so a(3,1)=1.
There are 3 distinct 3 X 3 matrices containing a 1 and otherwise 0's, so a(3,2)=3.
There are 12 distinct 3 X 3 matrices containing a single 1, a single 2 and otherwise 0's, so a(3,3)=12.
There is a single distinct 3 X 3 matrix containing all zeros, so a(3, 0) = 1.
There are 3 distinct 3 X 3 matrices containing 8 0's and a 1, so a(3, 1) = 3.
There are 12 distinct 3 X 3 matrices containing a single 1, a single 2 and otherwise 0's, so a(3, 2) = 12.
		

Links

• Eric Weisstein's World of Mathematics, Array

Crossrefs

Cf. A086829, A085698.

Programs

Mathematica

(For 3 X 3 case) CanonicalizeArray[ x_ ] := Module[ {r, t}, Sort[ {x, Reverse[ x ], r=Reverse/@x, Reverse[ r ], t=Transpose[ x ], Reverse[ t ], r=Reverse/@t, Reverse[ r ]} ][ [ 1 ] ] ] ADD[ n_, d_ ] := Union[ CanonicalizeArray /@ (Partition[ #, n ] & /@ Permutations[ Join[ Range[ d ], Table[ 0, {n^2 - d} ] ] ]) ]

Formula

T(n, k) = 1/8*([n^2]k+2*[n]_k+5*[0]_k) if n is even and 1/8*([n^2]_k+4*[n]_k+3*[1]_k) if n is odd, where [m]_k=m*(m-1)*...*(m-k+1), k>0, [m]_0=1, is falling factorial. - _Vladeta Jovovic, Aug 10 2003

Extensions

More terms from David Wasserman, Apr 12 2005