cp's OEIS Frontend

A242106 Number T(n,k) of inequivalent n X n matrices using exactly k different symbols, where equivalence means permutations of rows or columns or the symbol set; triangle T(n,k), n>=0, 0<=k<=n^2, read by rows.

%I A242106 #33 Feb 15 2023 19:58:38
%S A242106 1,0,1,0,1,4,3,1,0,1,17,121,269,241,100,24,3,1,0,1,172,15239,316622,
%T A242106 1951089,4820228,5769214,3768929,1451594,347251,53628,5645,451,37,3,1,
%U A242106 0,1,2811,10802952,3316523460,170309112972,2577666563670,15839885888526
%N A242106 Number T(n,k) of inequivalent n X n matrices using exactly k different symbols, where equivalence means permutations of rows or columns or the symbol set; triangle T(n,k), n>=0, 0<=k<=n^2, read by rows.
%C A242106 Note that the sequence with very similar number A246106 is related but different! - _M. F. Hasler_, Apr 29 2022
%H A242106 Alois P. Heinz, <a href="/A242106/b242106.txt">Rows n = 0..6, flattened</a>
%F A242106 T(n,k) = A242095(n,k) - A242095(n,k-1) for k>0. T(n,0) = A242095(n,0).
%e A242106 T(2,2) = 4:
%e A242106   [1 0]  [1 1]  [1 0]  [1 0]
%e A242106   [0 0], [0 0], [1 0], [0 1].
%e A242106 Triangle T(n,k) begins:
%e A242106   1;
%e A242106   0, 1;
%e A242106   0, 1, 4, 3, 1;
%e A242106   0, 1, 17, 121, 269, 241, 100, 24, 3, 1;
%e A242106   0, 1, 172, 15239, 316622, 1951089, 4820228, 5769214, 3768929, ...
%e A242106   0, 1, 2811, 10802952, 3316523460, 170309112972, 2577666563670, ...
%e A242106   0, 1, 126445, 50459558944, 382379913244053, 233995925116415261, ...
%p A242106 with(numtheory):
%p A242106 b:= proc(n, i) option remember; `if`(n=0, {0}, `if`(i<1, {},
%p A242106       {seq(map(p-> p+j*x^i, b(n-i*j, i-1) )[], j=0..n/i)}))
%p A242106     end:
%p A242106 A:= proc(n, k) option remember; add(add(add(mul(mul(add(d*
%p A242106       coeff(u, x, d), d=divisors(ilcm(i, j)))^(igcd(i, j)*
%p A242106       coeff(s, x, i)*coeff(t, x, j)), j=1..degree(t)),
%p A242106       i=1..degree(s))/mul(i^coeff(u, x, i)*coeff(u, x, i)!,
%p A242106       i=1..degree(u))/mul(i^coeff(t, x, i)*coeff(t, x, i)!,
%p A242106       i=1..degree(t))/mul(i^coeff(s, x, i)*coeff(s, x, i)!,
%p A242106       i=1..degree(s)), u=b(k$2)), t=b(n$2)), s=b(n$2))
%p A242106     end:
%p A242106 T:= (n, k)-> A(n, k) -A(n, k-1):
%p A242106 seq(seq(T(n, k), k=0..n^2), n=0..4);
%t A242106 Unprotect[Power]; 0^0 = 1; Protect[Power]; b[n_, i_] := b[n, i] = If[n == 0, {0}, If[i<1, {}, Table[Map[Function[{p}, p+j*x^i], b[n-i*j, i-1]], {j, 0, n/i}] // Flatten]]; A[n_, k_] := A[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[k, k]}], {t, b[n, n]}], {s, b[n, n]}]; T[n_, k_] := A[n, k] - A[n, k-1]; Table[ Table[T[n, k], {k, 0, n^2}], {n, 0, 4}] // Flatten (* _Jean-François Alcover_, Feb 13 2015, after _Alois P. Heinz_ *)
%Y A242106 Row sums give A091057.
%Y A242106 Main diagonal gives A360664.
%Y A242106 Cf. A242095, A246106.
%K A242106 nonn,tabf,look
%O A242106 0,6
%A A242106 _Alois P. Heinz_, Aug 15 2014