A242106 Number T(n,k) of inequivalent n X n matrices using exactly k different symbols, where equivalence means permutations of rows or columns or the symbol set; triangle T(n,k), n>=0, 0<=k<=n^2, read by rows.

1, 0, 1, 0, 1, 4, 3, 1, 0, 1, 17, 121, 269, 241, 100, 24, 3, 1, 0, 1, 172, 15239, 316622, 1951089, 4820228, 5769214, 3768929, 1451594, 347251, 53628, 5645, 451, 37, 3, 1, 0, 1, 2811, 10802952, 3316523460, 170309112972, 2577666563670, 15839885888526

Offset: 0

Author

Alois P. Heinz, Aug 15 2014

Keywords

• nonn
• tabf
• look

Comments

Note that the sequence with very similar number A246106 is related but different! - M. F. Hasler, Apr 29 2022

Examples

			T(2,2) = 4:
  [1 0]  [1 1]  [1 0]  [1 0]
  [0 0], [0 0], [1 0], [0 1].
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1, 4, 3, 1;
  0, 1, 17, 121, 269, 241, 100, 24, 3, 1;
  0, 1, 172, 15239, 316622, 1951089, 4820228, 5769214, 3768929, ...
  0, 1, 2811, 10802952, 3316523460, 170309112972, 2577666563670, ...
  0, 1, 126445, 50459558944, 382379913244053, 233995925116415261, ...
		

Links

• Alois P. Heinz, Rows n = 0..6, flattened

Crossrefs

Row sums give A091057.

Main diagonal gives A360664.

Cf. A242095, 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:
T:= (n, k)-> A(n, k) -A(n, k-1):
seq(seq(T(n, k), k=0..n^2), n=0..4);

Mathematica

Unprotect[Power]; 0^0 = 1; Protect[Power]; b[n_, i_] := b[n, i] = If[n == 0, {0}, If[i<1, {}, Table[Map[Function[{p}, p+j*x^i], b[n-i*j, i-1]], {j, 0, n/i}] // Flatten]]; A[n_, k_] := A[n, k] = Sum[ Sum[ Sum[ Product[ Product[ Sum[d* Coefficient[u, x, d], {d, Divisors[LCM[i, j]]}]^(GCD[i, j]*Coefficient[s, x, i] * Coefficient[t, x, j]), {j, 1, Exponent[t, x]}], {i, 1, Exponent[s, x]}] / Product[ i^Coefficient[u, x, i]*Coefficient[u, x, i]!, {i, 1, Exponent[u, x]}] / Product[ i^Coefficient[t, x, i]*Coefficient[t, x, i]!, {i, 1, Exponent[t, x]}] / Product[ i^Coefficient[s, x, i]*Coefficient[s, x, i]!, {i, 1, Exponent[s, x]}], {u, b[k, k]}], {t, b[n, n]}], {s, b[n, n]}]; T[n_, k_] := A[n, k] - A[n, k-1]; Table[ Table[T[n, k], {k, 0, n^2}], {n, 0, 4}] // Flatten (* Jean-François Alcover, Feb 13 2015, after Alois P. Heinz *)

Formula

T(n,k) = A242095(n,k) - A242095(n,k-1) for k>0. T(n,0) = A242095(n,0).