A256069 Number T(n,k) of inequivalent n X n matrices with entry set {1,...,k}, where equivalence means permutations of rows or columns; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 1, 5, 0, 1, 34, 633, 0, 1, 315, 89544, 7520386, 0, 1, 5622, 64780113, 79587235420, 20435529209470, 0, 1, 251608, 302752112913, 9177112514843320, 28079504654455279395, 19740907671252532135134

Offset: 0

Author

Alois P. Heinz, Mar 13 2015

Keywords

• nonn
• tabl

Examples

			T(2,2) = 5:
  [1 1]  [1 2]  [1 2]  [1 1]  [1 2]
  [1 2]  [2 2]  [1 2]  [2 2]  [2 1].
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,    5;
  0, 1,   34,      633;
  0, 1,  315,    89544,     7520386;
  0, 1, 5622, 64780113, 79587235420, 20435529209470;
		

Links

• Alois P. Heinz, Rows n = 0..20, flattened
• Index to number of inequivalent matrices modulo permutation of row and columns

Crossrefs

Cf. A246106.

Main diagonal gives A256070.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, [[]],
      `if`(i<1, [], [b(n, i-1)[], seq(map(p->[p[], [i, j]],
       b(n-i*j, i-1))[], j=1..n/i)]))
    end:
A:= proc(n, k) option remember; add(add(k^add(add(i[2]*j[2]*
      igcd(i[1], j[1]), j=t), i=s) /mul(i[1]^i[2]*i[2]!, i=s)
      /mul(i[1]^i[2]*i[2]!, i=t), t=b(n$2)), s=b(n$2))
    end:
T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..8);

Formula

T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A246106(n,k-i).