A242106 -id:A242106 - cp's OEIS frontend

A246106 Number A(n,k) of inequivalent n X n matrices with entries from [k], where equivalence means permutations of rows or columns; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 7, 1, 0, 1, 4, 27, 36, 1, 0, 1, 5, 76, 738, 317, 1, 0, 1, 6, 175, 8240, 90492, 5624, 1, 0, 1, 7, 351, 57675, 7880456, 64796982, 251610, 1, 0, 1, 8, 637, 289716, 270656150, 79846389608, 302752867740, 33642660, 1, 0

Offset: 0

Alois P. Heinz, Aug 13 2014

			Square array A(n,k) begins:
  1, 1,    1,        1,           1,              1, ...
  0, 1,    2,        3,           4,              5, ...
  0, 1,    7,       27,          76,            175, ...
  0, 1,   36,      738,        8240,          57675, ...
  0, 1,  317,    90492,     7880456,      270656150, ...
  0, 1, 5624, 64796982, 79846389608, 20834113243925, ...

Columns k = 0-10 give: A000007, A000012, A002724, A052269, A052271, A052272, A246112, A246113, A246114, A246115, A246116.
Rows n = 0-10 give: A000012, A001477, A039623, A058001, A058002, A058003, A058004, A246108, A246109, A246110, A246111.
Main diagonal gives A246107.
A028657, A242106, A353585 are related tables.
Cf. A242095, A256069.

b:= proc(n, i) option remember; `if`(n=0, [[]],
      `if`(i<1, [], [b(n, i-1)[], seq(map(p->[p[], [i, j]],
       b(n-i*j, i-1))[], j=1..n/i)]))
    end:
A:= proc(n, k) option remember; add(add(k^add(add(i[2]*j[2]*
      igcd(i[1], j[1]), j=t), i=s) /mul(i[1]^i[2]*i[2]!, i=s)
      /mul(i[1]^i[2]*i[2]!, i=t), t=b(n$2)), s=b(n$2))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..10);

A246106(n,k)=A353585(k,n,n) \\ M. F. Hasler, May 01 2022

A(n,k) = Sum_{i=0..k} C(k,i) * A256069(n,i).

A(n,k) = Sum_{p,q in P(n)} k^Sum_{i in p, j in q} gcd(i, j) / (N(p)*N(q)) where N(p) = Product_{distinct parts x in p} x^m(x)*m(x)!, m(x) = multiplicity of x in p. - M. F. Hasler, Apr 30 2022 [corrected by Anders Kaseorg, Oct 04 2024]

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.

Original entry on oeis.org

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

Alois P. Heinz, Aug 14 2014

A(n,k) = A(n,k+1) for k >= n^2.

			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, ...

Columns k=0-10 give: A000007, A000012, A091059, A091060, A091061, A091062, A246122, A246123, A246124, A246125, A246126.
Main diagonal gives A091058.
A(n,n^2) gives A091057.
Cf. A242093, A242106, 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:
seq(seq(A(n, d-n), n=0..d), d=0..10);

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 *)

A091057 Number of n X n matrices over symbol set {1,...,n^2} equivalent under any permutation of row, columns or the symbol set.

Original entry on oeis.org

1, 1, 9, 777, 18500104, 322286625959257, 7368376339801908226685191, 422262377369187686156418513093399998333, 105882936532098986759153041871810253870024776751177723954

Offset: 0

Christian G. Bower, Dec 17 2003

nonn

C. G. Bower, Explanation of A091057-A091062

Cf. A091058-A091062, A242095.

Row sums of A242106.

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}]]];
A242095[n_, k_] := A242095[n, k] = With[{co = Coefficient, ex = Exponent}, Sum[Sum[Sum[Product[Product[With[{g = GCD[i, j]*co[s, x, i]*co[t, x, j]}, If[g == 0, 1, Sum[d*co[u, x, d], {d, Divisors[LCM[i, j]]}]^g]], {j, ex[t, x]}], {i, ex[s, x]}]/Product[i^co[u, x, i]*co[u, x, i]!, {i, ex[u, x]}]/Product[i^co[t, x, i]*co[t, x, i]!, {i, ex[t, x]}]/Product[i^co[s, x, i]*co[s, x, i]!, {i, ex[s, x]}], {u, b[k, k]}], {t, b[n, n]}], {s, b[n, n]}]];
a[n_] := A242095[n, n^2];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 6}] (* Jean-François Alcover, May 29 2023, after Alois P. Heinz in A242095 *)

a(n) = Sum_{1*s_1+2*s_2+...=n, 1*t_1+2*t_2+...=n, 1*u_1+2*u_2+...=n^2} (fixA[s_1, s_2, ...; t_1, t_2, ...; u_1, u_2, ...]/ (1^s_1*s_1!*2^s_2*s_2!*...*1^t_1*t_1!*2^t_2*t_2!*...*1^u_1*u_1!*2^u_2*u_2!*...)) where fixA[...] = Product_{i,j>=1} ((Sum_{d|lcm(i, j)} (d*u_d))^(gcd(i, j)*s_i*t_j)). - corrected by Max Alekseyev, Jun 03 2023

a(7), a(8) from Max Alekseyev, Feb 09 2010

A360664 Number of inequivalent n X n matrices using exactly n different symbols, where equivalence means permutations of rows or columns or the symbol set.

Original entry on oeis.org

1, 1, 4, 121, 316622, 170309112972, 27417944542834007012, 1999576637456562016308833727820, 95614444589289128555388037722252407519896044, 4112190260012069813083100063825969898935642267505877920574871

Offset: 0

Alois P. Heinz, Feb 15 2023

nonn

			a(2) = 4:
  [1 0]  [1 1]  [1 0]  [1 0]
  [0 0], [0 0], [1 0], [0 1].

Alois P. Heinz, Table of n, a(n) for n = 0..14

Main diagonal of A242106.

a(n) = A242106(n,n).

cp's OEIS Frontend

A246106 Number A(n,k) of inequivalent n X n matrices with entries from [k], where equivalence means permutations of rows or columns; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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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.

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A091057 Number of n X n matrices over symbol set {1,...,n^2} equivalent under any permutation of row, columns or the symbol set.

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A360664 Number of inequivalent n X n matrices using exactly n different symbols, where equivalence means permutations of rows or columns or the symbol set.

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