OFFSET
0,5
COMMENTS
In other words, two matrices are considered equivalent if one can be obtained from the other by some sequence of interchanges of the rows.
LINKS
Alois P. Heinz, Rows n = 0..32, flattened
EXAMPLE
T(2,2) = 4 because we have: {{0,0},{1,1}}; {{0,1},{1,0}}; {{0,1},{0,1}}; {{1,0},{1,0}} (where the first two matrices were arbitrarily selected as class representatives).
Triangle T(n,k) begins:
1;
1, 1;
1, 2, 4, 2, 1;
1, 3, 9, 20, 27, 27, 20, 9, 3, 1;
1, 4, 16, 48, 133, 272, 468, 636, 720, 636, 468, 272, 133, 48, 16, 4, 1;
...
MAPLE
g:= proc(n, i, j) option remember; expand(`if`(j=0, 1, `if`(i<0, 0, add(
g(n, i-1, j-k)*x^(i*k)*binomial(binomial(n, i)+k-1, k), k=0..j))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(g(n$3)):
seq(T(n), n=0..5); # Alois P. Heinz, Feb 15 2023
MATHEMATICA
nn=100; Table[CoefficientList[Series[CycleIndex[SymmetricGroup[n], s]/.Table[s[i]->(1+x^i)^n, {i, 1, n}], {x, 0, nn}], x], {n, 0, 5}]//Grid
(* Second program: *)
g[n_, i_, j_] := g[n, i, j] = Expand[If[j == 0, 1, If[i < 0, 0, Sum[g[n, i - 1, j - k]*x^(i*k)*Binomial[Binomial[n, i] + k - 1, k], {k, 0, j}]]]];
T[n_] := CoefficientList[g[n, n, n], x];
Table[T[n], {n, 0, 5}] // Flatten (* Jean-François Alcover, May 28 2023, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Geoffrey Critzer, Feb 20 2013
STATUS
approved