A242095 Number A(n,k) of inequivalent n X n matrices with entries from [k], where equivalence means permutations of rows or columns or the symbol set; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 5, 1, 0, 1, 1, 8, 18, 1, 0, 1, 1, 9, 139, 173, 1, 0, 1, 1, 9, 408, 15412, 2812, 1, 0, 1, 1, 9, 649, 332034, 10805764, 126446, 1, 0, 1, 1, 9, 749, 2283123, 3327329224, 50459685390, 16821330, 1, 0

Offset: 0

Author

Alois P. Heinz, Aug 14 2014

Keywords

• nonn
• tabl

Comments

A(n,k) = A(n,k+1) for k >= n^2.

Examples

			A(2,2) = 5:
  [1 1]  [2 1]  [2 2]  [2 1]  [2 1]
  [1 1], [1 1], [1 1], [2 1], [1 2].
Square array A(n,k) begins:
  1, 1,    1,        1,          1,            1, ...
  0, 1,    1,        1,          1,            1, ...
  0, 1,    5,        8,          9,            9, ...
  0, 1,   18,      139,        408,          649, ...
  0, 1,  173,    15412,     332034,      2283123, ...
  0, 1, 2812, 10805764, 3327329224, 173636442196, ...
		

Links

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

Crossrefs

Columns k=0-10 give: A000007, A000012, A091059, A091060, A091061, A091062, A246122, A246123, A246124, A246125, A246126.

Main diagonal gives A091058.

A(n,n^2) gives A091057.

Cf. A242093, A242106, A246106.

Programs

Maple

with(numtheory):
b:= proc(n, i) option remember; `if`(n=0, {0}, `if`(i<1, {},
      {seq(map(p-> p+j*x^i, b(n-i*j, i-1) )[], j=0..n/i)}))
    end:
A:= proc(n, k) option remember; add(add(add(mul(mul(add(d*
      coeff(u, x, d), d=divisors(ilcm(i, j)))^(igcd(i, j)*
      coeff(s, x, i)*coeff(t, x, j)), j=1..degree(t)),
      i=1..degree(s))/mul(i^coeff(u, x, i)*coeff(u, x, i)!,
      i=1..degree(u))/mul(i^coeff(t, x, i)*coeff(t, x, i)!,
      i=1..degree(t))/mul(i^coeff(s, x, i)*coeff(s, x, i)!,
      i=1..degree(s)), u=b(k$2)), t=b(n$2)), s=b(n$2))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..10);

Mathematica

b[n_, i_] := b[n, i] = If[n==0, {0}, If[i<1, {}, Flatten@Table[Map[ Function[p, p + j*x^i], b[n - i*j, i - 1]], {j, 0, n/i}]]];
A[n_, k_] := A[n, k] = Sum[Sum[Sum[Product[Product[With[{g = GCD[i, j]* Coefficient[s, x, i]*Coefficient[t, x, j]}, If[g == 0, 1, Sum[d* Coefficient[u, x, d], {d, Divisors[LCM[i, j]]}]^g]], {j, Exponent[t, x]} ],
{i, Exponent[s, x]}]/Product[i^Coefficient[u, x, i]*Coefficient[u, x, i]!,
{i, Exponent[u, x]}]/Product[i^Coefficient[t, x, i]*Coefficient[t, x, i]!,
{i, Exponent[t, x]}]/Product[i^Coefficient[s, x, i]*Coefficient[s, x, i]!,
{i, Exponent[s, x]}], {u, b[k, k]}], {t, b[n, n]}], {s, b[n, n]}];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Feb 21 2016, after Alois P. Heinz, updated Jan 01 2021 *)