A091057 Number of n X n matrices over symbol set {1,...,n^2} equivalent under any permutation of row, columns or the symbol set.

1, 1, 9, 777, 18500104, 322286625959257, 7368376339801908226685191, 422262377369187686156418513093399998333, 105882936532098986759153041871810253870024776751177723954

Offset

0,3

Links

Table of n, a(n) for n=0..8.

C. G. Bower, Explanation of A091057-A091062

Formula

a(n) = Sum_{1*s_1+2*s_2+...=n, 1*t_1+2*t_2+...=n, 1*u_1+2*u_2+...=n^2} (fixA[s_1, s_2, ...; t_1, t_2, ...; u_1, u_2, ...]/ (1^s_1*s_1!*2^s_2*s_2!*...*1^t_1*t_1!*2^t_2*t_2!*...*1^u_1*u_1!*2^u_2*u_2!*...)) where fixA[...] = Product_{i,j>=1} ((Sum_{d|lcm(i, j)} (d*u_d))^(gcd(i, j)*s_i*t_j)). - corrected by Max Alekseyev, Jun 03 2023

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}]]];
A242095[n_, k_] := A242095[n, k] = With[{co = Coefficient, ex = Exponent}, Sum[Sum[Sum[Product[Product[With[{g = GCD[i, j]*co[s, x, i]*co[t, x, j]}, If[g == 0, 1, Sum[d*co[u, x, d], {d, Divisors[LCM[i, j]]}]^g]], {j, ex[t, x]}], {i, ex[s, x]}]/Product[i^co[u, x, i]*co[u, x, i]!, {i, ex[u, x]}]/Product[i^co[t, x, i]*co[t, x, i]!, {i, ex[t, x]}]/Product[i^co[s, x, i]*co[s, x, i]!, {i, ex[s, x]}], {u, b[k, k]}], {t, b[n, n]}], {s, b[n, n]}]];
a[n_] := A242095[n, n^2];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 6}] (* Jean-François Alcover, May 29 2023, after Alois P. Heinz in A242095 *)

Crossrefs

Cf. A091058-A091062, A242095.

Row sums of A242106.

Sequence in context: A223380 A246125 A246126 * A168257 A303143 A137065

Adjacent sequences: A091054 A091055 A091056 * A091058 A091059 A091060

Keyword

nonn

Author

Christian G. Bower, Dec 17 2003

Extensions

a(7), a(8) from Max Alekseyev, Feb 09 2010

Status

approved