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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.

%I A242095 #32 May 02 2022 16:00:43
%S A242095 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,
%T A242095 408,15412,2812,1,0,1,1,9,649,332034,10805764,126446,1,0,1,1,9,749,
%U A242095 2283123,3327329224,50459685390,16821330,1,0
%N 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.
%C A242095 A(n,k) = A(n,k+1) for k >= n^2.
%H A242095 Alois P. Heinz, <a href="/A242095/b242095.txt">Antidiagonals n = 0..20, flattened</a>
%H A242095 <a href="/index/Mat#inequiv">Index to number of inequivalent matrices modulo permutation of rows and columns</a>
%e A242095 A(2,2) = 5:
%e A242095   [1 1]  [2 1]  [2 2]  [2 1]  [2 1]
%e A242095   [1 1], [1 1], [1 1], [2 1], [1 2].
%e A242095 Square array A(n,k) begins:
%e A242095   1, 1,    1,        1,          1,            1, ...
%e A242095   0, 1,    1,        1,          1,            1, ...
%e A242095   0, 1,    5,        8,          9,            9, ...
%e A242095   0, 1,   18,      139,        408,          649, ...
%e A242095   0, 1,  173,    15412,     332034,      2283123, ...
%e A242095   0, 1, 2812, 10805764, 3327329224, 173636442196, ...
%p A242095 with(numtheory):
%p A242095 b:= proc(n, i) option remember; `if`(n=0, {0}, `if`(i<1, {},
%p A242095       {seq(map(p-> p+j*x^i, b(n-i*j, i-1) )[], j=0..n/i)}))
%p A242095     end:
%p A242095 A:= proc(n, k) option remember; add(add(add(mul(mul(add(d*
%p A242095       coeff(u, x, d), d=divisors(ilcm(i, j)))^(igcd(i, j)*
%p A242095       coeff(s, x, i)*coeff(t, x, j)), j=1..degree(t)),
%p A242095       i=1..degree(s))/mul(i^coeff(u, x, i)*coeff(u, x, i)!,
%p A242095       i=1..degree(u))/mul(i^coeff(t, x, i)*coeff(t, x, i)!,
%p A242095       i=1..degree(t))/mul(i^coeff(s, x, i)*coeff(s, x, i)!,
%p A242095       i=1..degree(s)), u=b(k$2)), t=b(n$2)), s=b(n$2))
%p A242095     end:
%p A242095 seq(seq(A(n, d-n), n=0..d), d=0..10);
%t A242095 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 A242095 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]} ],
%t A242095 {i, Exponent[s, x]}]/Product[i^Coefficient[u, x, i]*Coefficient[u, x, i]!,
%t A242095 {i, Exponent[u, x]}]/Product[i^Coefficient[t, x, i]*Coefficient[t, x, i]!,
%t A242095 {i, Exponent[t, x]}]/Product[i^Coefficient[s, x, i]*Coefficient[s, x, i]!,
%t A242095 {i, Exponent[s, x]}], {u, b[k, k]}], {t, b[n, n]}], {s, b[n, n]}];
%t A242095 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 *)
%Y A242095 Columns k=0-10 give: A000007, A000012, A091059, A091060, A091061, A091062, A246122, A246123, A246124, A246125, A246126.
%Y A242095 Main diagonal gives A091058.
%Y A242095 A(n,n^2) gives A091057.
%Y A242095 Cf. A242093, A242106, A246106.
%K A242095 nonn,tabl
%O A242095 0,13
%A A242095 _Alois P. Heinz_, Aug 14 2014