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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.
15
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
OFFSET
0,13
COMMENTS
A(n,k) = A(n,k+1) for k >= n^2.
EXAMPLE
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, ...
MAPLE
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);
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}]]];
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 *)
CROSSREFS
Main diagonal gives A091058.
A(n,n^2) gives A091057.
Sequence in context: A281563 A293087 A209349 * A336169 A007912 A019755
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Aug 14 2014
STATUS
approved