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A256069 Number T(n,k) of inequivalent n X n matrices with entry set {1,...,k}, where equivalence means permutations of rows or columns; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%I A256069 #13 May 02 2022 09:32:55
%S A256069 1,0,1,0,1,5,0,1,34,633,0,1,315,89544,7520386,0,1,5622,64780113,
%T A256069 79587235420,20435529209470,0,1,251608,302752112913,9177112514843320,
%U A256069 28079504654455279395,19740907671252532135134
%N A256069 Number T(n,k) of inequivalent n X n matrices with entry set {1,...,k}, where equivalence means permutations of rows or columns; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%H A256069 Alois P. Heinz, <a href="/A256069/b256069.txt">Rows n = 0..20, flattened</a>
%H A256069 <a href="/index/Mat#inequiv">Index to number of inequivalent matrices modulo permutation of row and columns</a>
%F A256069 T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A246106(n,k-i).
%e A256069 T(2,2) = 5:
%e A256069   [1 1]  [1 2]  [1 2]  [1 1]  [1 2]
%e A256069   [1 2]  [2 2]  [1 2]  [2 2]  [2 1].
%e A256069 Triangle T(n,k) begins:
%e A256069   1;
%e A256069   0, 1;
%e A256069   0, 1,    5;
%e A256069   0, 1,   34,      633;
%e A256069   0, 1,  315,    89544,     7520386;
%e A256069   0, 1, 5622, 64780113, 79587235420, 20435529209470;
%p A256069 b:= proc(n, i) option remember; `if`(n=0, [[]],
%p A256069       `if`(i<1, [], [b(n, i-1)[], seq(map(p->[p[], [i, j]],
%p A256069        b(n-i*j, i-1))[], j=1..n/i)]))
%p A256069     end:
%p A256069 A:= proc(n, k) option remember; add(add(k^add(add(i[2]*j[2]*
%p A256069       igcd(i[1], j[1]), j=t), i=s) /mul(i[1]^i[2]*i[2]!, i=s)
%p A256069       /mul(i[1]^i[2]*i[2]!, i=t), t=b(n$2)), s=b(n$2))
%p A256069     end:
%p A256069 T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
%p A256069 seq(seq(T(n, k), k=0..n), n=0..8);
%Y A256069 Cf. A246106.
%Y A256069 Main diagonal gives A256070.
%K A256069 nonn,tabl
%O A256069 0,6
%A A256069 _Alois P. Heinz_, Mar 13 2015