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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

%I #32 May 02 2022 16:00:43

%S 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,

%T 408,15412,2812,1,0,1,1,9,649,332034,10805764,126446,1,0,1,1,9,749,

%U 2283123,3327329224,50459685390,16821330,1,0

%N 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.

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

%H Alois P. Heinz, <a href="/A242095/b242095.txt">Antidiagonals n = 0..20, flattened</a>

%H <a href="/index/Mat#inequiv">Index to number of inequivalent matrices modulo permutation of rows and columns</a>

%e A(2,2) = 5:

%e [1 1] [2 1] [2 2] [2 1] [2 1]

%e [1 1], [1 1], [1 1], [2 1], [1 2].

%e Square array A(n,k) begins:

%e 1, 1, 1, 1, 1, 1, ...

%e 0, 1, 1, 1, 1, 1, ...

%e 0, 1, 5, 8, 9, 9, ...

%e 0, 1, 18, 139, 408, 649, ...

%e 0, 1, 173, 15412, 332034, 2283123, ...

%e 0, 1, 2812, 10805764, 3327329224, 173636442196, ...

%p with(numtheory):

%p b:= proc(n, i) option remember; `if`(n=0, {0}, `if`(i<1, {},

%p {seq(map(p-> p+j*x^i, b(n-i*j, i-1) )[], j=0..n/i)}))

%p end:

%p A:= proc(n, k) option remember; add(add(add(mul(mul(add(d*

%p coeff(u, x, d), d=divisors(ilcm(i, j)))^(igcd(i, j)*

%p coeff(s, x, i)*coeff(t, x, j)), j=1..degree(t)),

%p i=1..degree(s))/mul(i^coeff(u, x, i)*coeff(u, x, i)!,

%p i=1..degree(u))/mul(i^coeff(t, x, i)*coeff(t, x, i)!,

%p i=1..degree(t))/mul(i^coeff(s, x, i)*coeff(s, x, i)!,

%p i=1..degree(s)), u=b(k$2)), t=b(n$2)), s=b(n$2))

%p end:

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

%t 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}]]];

%t 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]} ],

%t {i, Exponent[s, x]}]/Product[i^Coefficient[u, x, i]*Coefficient[u, x, i]!,

%t {i, Exponent[u, x]}]/Product[i^Coefficient[t, x, i]*Coefficient[t, x, i]!,

%t {i, Exponent[t, x]}]/Product[i^Coefficient[s, x, i]*Coefficient[s, x, i]!,

%t {i, Exponent[s, x]}], {u, b[k, k]}], {t, b[n, n]}], {s, b[n, n]}];

%t 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 *)

%Y Columns k=0-10 give: A000007, A000012, A091059, A091060, A091061, A091062, A246122, A246123, A246124, A246125, A246126.

%Y Main diagonal gives A091058.

%Y A(n,n^2) gives A091057.

%Y Cf. A242093, A242106, A246106.

%K nonn,tabl

%O 0,13

%A _Alois P. Heinz_, Aug 14 2014

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Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)