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A242093 Number A(n,k) of inequivalent n X k binary matrices, where equivalence means permutations of rows or columns or the symbol set; square array A(n,k), n>=0, k>=0, read by antidiagonals. 13
1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 5, 2, 1, 1, 3, 8, 8, 3, 1, 1, 3, 14, 18, 14, 3, 1, 1, 4, 20, 47, 47, 20, 4, 1, 1, 4, 30, 95, 173, 95, 30, 4, 1, 1, 5, 40, 200, 545, 545, 200, 40, 5, 1, 1, 5, 55, 367, 1682, 2812, 1682, 367, 55, 5, 1, 1, 6, 70, 674, 4745, 14386, 14386, 4745, 674, 70, 6, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

LINKS

Alois P. Heinz, Antidiagonals n = 0..35, flattened

EXAMPLE

A(1,4) = 3: [0 0 0 0], [1 0 0 0], [1 1 0 0].

A(1,5) = 3: [0 0 0 0 0], [1 0 0 0 0], [1 1 0 0 0].

A(2,2) = 5:

  [0 0]  [1 0]  [1 1]  [1 0]  [1 0]

  [0 0], [0 0], [0 0], [1 0], [0 1].

A(3,2) = 8:

  [0 0]  [1 0]  [1 1]  [1 0]  [1 0]  [1 0]  [1 0]  [1 1]

  [0 0], [0 0], [0 0], [1 0], [0 1], [1 0], [0 1], [1 0].

  [0 0]  [0 0]  [0 0]  [0 0]  [0 0]  [1 0]  [1 0]  [0 0]

Square array A(n,k) begins:

  1, 1,  1,   1,    1,     1,       1,        1, ...

  1, 1,  2,   2,    3,     3,       4,        4, ...

  1, 2,  5,   8,   14,    20,      30,       40, ...

  1, 2,  8,  18,   47,    95,     200,      367, ...

  1, 3, 14,  47,  173,   545,    1682,     4745, ...

  1, 3, 20,  95,  545,  2812,   14386,    68379, ...

  1, 4, 30, 200, 1682, 14386,  126446,  1072086, ...

  1, 4, 40, 367, 4745, 68379, 1072086, 16821330, ...

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:

g:= 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(2$2)), t=b(n$2)), s=b(k$2))

    end:

A:= (n, k)-> g(sort([n, k])[]):

seq(seq(A(n, d-n), n=0..d), d=0..12);

MATHEMATICA

Unprotect[Power]; 0^0 = 1;

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}]]]];

g[n_, k_] := g[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[2, 2]}], {t, b[n, n]}], {s, b[k, k]}];

A[n_, k_] := g @@ Sort[{n, k}];

Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-Fran├žois Alcover, Apr 25 2016, adapted from Maple *)

CROSSREFS

Columns (or rows) k=0-10 give: A000012, A008619, A006918(n+1), A246148, A246149, A246150, A246151, A246152, A246153, A246154, A246155.

Main diagonal gives A091059.

Cf. A028657, A241956, A242095.

Sequence in context: A131373 A245185 A034853 * A322058 A244006 A110283

Adjacent sequences:  A242090 A242091 A242092 * A242094 A242095 A242096

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Aug 14 2014

STATUS

approved

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Last modified February 22 17:35 EST 2019. Contains 320400 sequences. (Running on oeis4.)