A348191 Triangular array read by rows: T(n,k) is the number of cubic n-permutations possessing exactly k cycles; n >= 0, 0 <= k <= n.

1, 0, 1, 0, 1, 1, 0, 0, 3, 1, 0, 6, 3, 6, 1, 0, 24, 30, 15, 10, 1, 0, 0, 234, 105, 45, 15, 1, 0, 720, 504, 1134, 315, 105, 21, 1, 0, 5040, 7020, 5292, 3969, 840, 210, 28, 1, 0, 0, 89424, 48572, 29484, 11529, 2016, 378, 36, 1, 0, 362880, 299376, 724140, 275120, 118125, 29673, 4410, 630, 45, 1

Offset: 0

Author

Steven Finch, Nov 27 2021

Keywords

• nonn
• tabl

Comments

A permutation p in S_n is a cube if there exists q in S_n with q^3=p.

Examples

			The four cubic 3-permutations are (1, 2, 3) with three cycles (fixed points) and (1, 3, 2), (3, 2, 1) & (2, 1, 3), each with two cycles (a fixed point & a transposition).
Triangle begins:
[0]  1;
[1]  0,   1;
[2]  0,   1,   1;
[3]  0,   0,   3,    1;
[4]  0,   6,   3,    6,   1;
[5]  0,  24,  30,   15,  10,   1;
[6]  0,   0, 234,  105,  45,  15,  1;
[7]  0, 720, 504, 1134, 315, 105, 21, 1;
		

Links

• Alois P. Heinz, Rows n = 0..150, flattened

Crossrefs

Columns k=0-1 give: A000007, |A194770|.

Row sums give A103619.

Cf. A246948.

Programs

Maple

with(combinat):
b:= proc(n, i) option remember; expand(`if`(n=0, 1, `if`(i<1, 0,
      add(`if`(irem(j, igcd(i, 3))<>0, 0, x^j*(i-1)!^j*
      multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1)), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
seq(T(n), n=0..10);  # Alois P. Heinz, Nov 30 2021

Mathematica

multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i<1, 0,
     Sum[If[Mod[j, GCD[i, 3]] != 0, 0, x^j*(i-1)!^j*multinomial[n,
     Join[{n-i*j}, Table[i, {j}]]]/j!*b[n-i*j, i-1]], {j, 0, n/i}]]]];
T[n_] := With[{p = b[n, n]}, Table[Coefficient[p, x, i], {i, 0, n}]];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Dec 28 2021, after Alois P. Heinz *)

Extensions

More terms from Alois P. Heinz, Nov 30 2021