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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. 15
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,13

COMMENTS

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

LINKS

Alois P. Heinz, Antidiagonals n = 0..20, flattened

EXAMPLE

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, ...

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:

seq(seq(A(n, d-n), n=0..d), d=0..10);

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}]]];

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-Fran├žois Alcover, Feb 21 2016, after Alois P. Heinz, updated Jan 01 2021 *)

CROSSREFS

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.

Sequence in context: A281563 A293087 A209349 * A336169 A007912 A019755

Adjacent sequences:  A242092 A242093 A242094 * A242096 A242097 A242098

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Aug 14 2014

STATUS

approved

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Last modified October 21 13:44 EDT 2021. Contains 348155 sequences. (Running on oeis4.)