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A242095
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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.
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15
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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, 408, 15412, 2812, 1, 0, 1, 1, 9, 649, 332034, 10805764, 126446, 1, 0, 1, 1, 9, 749, 2283123, 3327329224, 50459685390, 16821330, 1, 0
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OFFSET
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0,13
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COMMENTS
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A(n,k) = A(n,k+1) for k >= n^2.
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LINKS
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EXAMPLE
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A(2,2) = 5:
[1 1] [2 1] [2 2] [2 1] [2 1]
[1 1], [1 1], [1 1], [2 1], [1 2].
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, ...
0, 1, 1, 1, 1, 1, ...
0, 1, 5, 8, 9, 9, ...
0, 1, 18, 139, 408, 649, ...
0, 1, 173, 15412, 332034, 2283123, ...
0, 1, 2812, 10805764, 3327329224, 173636442196, ...
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MAPLE
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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:
seq(seq(A(n, d-n), n=0..d), d=0..10);
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MATHEMATICA
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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}]]];
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]} ],
{i, Exponent[s, x]}]/Product[i^Coefficient[u, x, i]*Coefficient[u, x, i]!,
{i, Exponent[u, x]}]/Product[i^Coefficient[t, x, i]*Coefficient[t, x, i]!,
{i, Exponent[t, x]}]/Product[i^Coefficient[s, x, i]*Coefficient[s, x, i]!,
{i, Exponent[s, x]}], {u, b[k, k]}], {t, b[n, n]}], {s, b[n, n]}];
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 *)
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CROSSREFS
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Columns k=0-10 give: A000007, A000012, A091059, A091060, A091061, A091062, A246122, A246123, A246124, A246125, A246126.
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KEYWORD
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AUTHOR
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STATUS
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approved
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