A360664 -id:A360664 - cp's OEIS frontend

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.

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

Alois P. Heinz, Aug 15 2014

Note that the sequence with very similar number A246106 is related but different! - M. F. Hasler, Apr 29 2022

			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, ...

Alois P. Heinz, Rows n = 0..6, flattened

Row sums give A091057.
Main diagonal gives A360664.
Cf. A242095, A246106.

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);

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 *)

T(n,k) = A242095(n,k) - A242095(n,k-1) for k>0. T(n,0) = A242095(n,0).

A360660 Number of inequivalent n X n {0,1} matrices modulo permutation of the rows, with exactly n 1's.

Original entry on oeis.org

1, 1, 4, 20, 133, 1027, 9259, 94033, 1062814, 13176444, 177427145, 2573224238, 39924120823, 658921572675, 11513293227271, 212109149134617, 4105637511110979, 83238756058333277, 1762856698153603049, 38905470655863251479, 892840913430059075405

Offset: 0

Alois P. Heinz, Feb 15 2023

nonn

Also the number of multisets of n words of length n over binary alphabet where the first letter occurs n times. E.g., a(2) = 4: {aa,bb}, {ab,ab}, {ab,ba}, {ba,ba}.

			a(3) = 20: [111/000/000], [110/100/000], [110/010/000], [110/001/000], [101/100/000], [101/010/000], [101/001/000], [100/100/100], [100/100/010], [100/100/001], [100/011/000], [100/010/010], [100/010/001], [100/001/001], [011/010/000], [011/001/000], [010/010/010], [010/010/001], [010/001/001], [001/001/001].

Alois P. Heinz, Table of n, a(n) for n = 0..200

Main diagonal of A220886.

Cf. A091058, A091059, A154323, A246107, A256070, A360664, A383073.

g:= proc(n, i, j) option remember; expand(`if`(j=0, 1, `if`(i<0, 0, add(
      g(n, i-1, j-k)*x^(i*k)*binomial(binomial(n, i)+k-1, k), k=0..j))))
    end:
a:= n-> coeff(g(n$3), x, n):
seq(a(n), n=0..20);

g[n_, i_, j_] := g[n, i, j] = Expand[If[j == 0, 1, If[i < 0, 0, Sum[g[n, i - 1, j - k]*x^(i*k)*Binomial[Binomial[n, i] + k - 1, k], {k, 0, j}]]]];
a[n_] := SeriesCoefficient[g[n, n, n], {x, 0, n}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, May 28 2023, after Alois P. Heinz *)
Table[SeriesCoefficient[Product[1/(1 - x^k)^Binomial[n, k], {k, 1, n}], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 15 2025 *)

a(n) = A220886(n,n).

a(n) = [x^n] Product_{k=1..n} 1/(1 - x^k)^binomial(n,k). - Vaclav Kotesovec, Apr 15 2025

cp's OEIS Frontend

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.

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