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

0,6

Links

Alois P. Heinz, Rows n = 0..20, flattened

Index to number of inequivalent matrices modulo permutation of row and columns

Formula

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

Example

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;

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);

Crossrefs

Cf. A246106.

Main diagonal gives A256070.

Sequence in context: A227322 A216718 A184180 * A356652 A267480 A099221

Adjacent sequences: A256066 A256067 A256068 * A256070 A256071 A256072

Keyword

nonn,tabl

Author

Alois P. Heinz, Mar 13 2015

Status

approved