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.
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
Offset: 0
Examples
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, ...
Links
Crossrefs
Programs
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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
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[With[{gc = GCD[i, j]* Coefficient[s, x, i]*Coefficient[t, x, j]}, If[gc == 0, 1, Sum[d* Coefficient[u, x, d], {d, Divisors[LCM[i, j]]}]^gc]], {j, 1, 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[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-François Alcover, Apr 25 2016, adapted from Maple, updated Jan 01 2021 *)