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A242106
Number T(n,k) of inequivalent n X n matrices using exactly k different symbols, where equivalence means permutations of rows or columns or the symbol set; triangle T(n,k), n>=0, 0<=k<=n^2, read by rows.
5
1, 0, 1, 0, 1, 4, 3, 1, 0, 1, 17, 121, 269, 241, 100, 24, 3, 1, 0, 1, 172, 15239, 316622, 1951089, 4820228, 5769214, 3768929, 1451594, 347251, 53628, 5645, 451, 37, 3, 1, 0, 1, 2811, 10802952, 3316523460, 170309112972, 2577666563670, 15839885888526
OFFSET
0,6
COMMENTS
Note that the sequence with very similar number A246106 is related but different! - M. F. Hasler, Apr 29 2022
LINKS
Alois P. Heinz, Rows n = 0..6, flattened
FORMULA
T(n,k) = A242095(n,k) - A242095(n,k-1) for k>0. T(n,0) = A242095(n,0).
EXAMPLE
T(2,2) = 4:
[1 0] [1 1] [1 0] [1 0]
[0 0], [0 0], [1 0], [0 1].
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 4, 3, 1;
0, 1, 17, 121, 269, 241, 100, 24, 3, 1;
0, 1, 172, 15239, 316622, 1951089, 4820228, 5769214, 3768929, ...
0, 1, 2811, 10802952, 3316523460, 170309112972, 2577666563670, ...
0, 1, 126445, 50459558944, 382379913244053, 233995925116415261, ...
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:
T:= (n, k)-> A(n, k) -A(n, k-1):
seq(seq(T(n, k), k=0..n^2), n=0..4);
MATHEMATICA
Unprotect[Power]; 0^0 = 1; Protect[Power]; b[n_, i_] := b[n, i] = If[n == 0, {0}, If[i<1, {}, Table[Map[Function[{p}, p+j*x^i], b[n-i*j, i-1]], {j, 0, n/i}] // Flatten]]; A[n_, k_] := A[n, k] = Sum[ Sum[ Sum[ Product[ Product[ Sum[d* Coefficient[u, x, d], {d, Divisors[LCM[i, j]]}]^(GCD[i, j]*Coefficient[s, x, i] * Coefficient[t, x, j]), {j, 1, Exponent[t, x]}], {i, 1, Exponent[s, x]}] / Product[ i^Coefficient[u, x, i]*Coefficient[u, x, i]!, {i, 1, Exponent[u, x]}] / Product[ i^Coefficient[t, x, i]*Coefficient[t, x, i]!, {i, 1, Exponent[t, x]}] / Product[ i^Coefficient[s, x, i]*Coefficient[s, x, i]!, {i, 1, Exponent[s, x]}], {u, b[k, k]}], {t, b[n, n]}], {s, b[n, n]}]; T[n_, k_] := A[n, k] - A[n, k-1]; Table[ Table[T[n, k], {k, 0, n^2}], {n, 0, 4}] // Flatten (* Jean-François Alcover, Feb 13 2015, after Alois P. Heinz *)
CROSSREFS
Row sums give A091057.
Main diagonal gives A360664.
Sequence in context: A054669 A131027 A133475 * A294885 A021236 A136590
KEYWORD
nonn,tabf,look
AUTHOR
Alois P. Heinz, Aug 15 2014
STATUS
approved