cp's OEIS Frontend

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

%I A091057 #28 Jun 04 2023 01:46:10
%S A091057 1,1,9,777,18500104,322286625959257,7368376339801908226685191,
%T A091057 422262377369187686156418513093399998333,
%U A091057 105882936532098986759153041871810253870024776751177723954
%N A091057 Number of n X n matrices over symbol set {1,...,n^2} equivalent under any permutation of row, columns or the symbol set.
%H A091057 C. G. Bower, <a href="/A091057/a091057.html">Explanation of A091057-A091062</a>
%F A091057 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
%t A091057 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}]]];
%t A091057 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]}]];
%t A091057 a[n_] := A242095[n, n^2];
%t A091057 Table[Print[n, " ", a[n]]; a[n], {n, 0, 6}] (* _Jean-François Alcover_, May 29 2023, after _Alois P. Heinz_ in A242095 *)
%Y A091057 Cf. A091058-A091062, A242095.
%Y A091057 Row sums of A242106.
%K A091057 nonn
%O A091057 0,3
%A A091057 _Christian G. Bower_, Dec 17 2003
%E A091057 a(7), a(8) from _Max Alekseyev_, Feb 09 2010