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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. 3
1, 0, 1, 0, 1, 4, 3, 1, 0, 1, 17, 121, 269, 241, 100, 24, 3, 1, 0, 1, 172, 15239, 316622, 1951089, 4820228, 5769214, 3768929, 1451594, 347251, 53628, 5645, 451, 37, 3, 1, 0, 1, 2811, 10802952, 3316523460, 170309112972, 2577666563670, 15839885888526 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

LINKS

Alois P. Heinz, Rows n = 0..6, flattened

FORMULA

T(n,k) = A242095(n,k) - A242095(n,k-1) for k>0. T(n,0) = A242095(n,0).

EXAMPLE

T(2,2) = 4:

  [1 0]  [1 1]  [1 0]  [1 0]

  [0 0], [0 0], [1 0], [0 1].

Triangle T(n,k) begins:

  1;

  0, 1;

  0, 1, 4, 3, 1;

  0, 1, 17, 121, 269, 241, 100, 24, 3, 1;

  0, 1, 172, 15239, 316622, 1951089, 4820228, 5769214, 3768929, ...

  0, 1, 2811, 10802952, 3316523460, 170309112972, 2577666563670, ...

  0, 1, 126445, 50459558944, 382379913244053, 233995925116415261, ...

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:

T:= (n, k)-> A(n, k) -A(n, k-1):

seq(seq(T(n, k), k=0..n^2), n=0..4);

MATHEMATICA

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-Fran├žois Alcover, Feb 13 2015, after Alois P. Heinz *)

CROSSREFS

Row sums give A091057.

Sequence in context: A054669 A131027 A133475 * A294885 A021236 A136590

Adjacent sequences:  A242103 A242104 A242105 * A242107 A242108 A242109

KEYWORD

nonn,tabf,look

AUTHOR

Alois P. Heinz, Aug 15 2014

STATUS

approved

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Last modified January 29 14:07 EST 2020. Contains 331338 sequences. (Running on oeis4.)