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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.

%I A348191 #22 Dec 28 2021 14:15:12
%S A348191 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,
%T A348191 720,504,1134,315,105,21,1,0,5040,7020,5292,3969,840,210,28,1,0,0,
%U A348191 89424,48572,29484,11529,2016,378,36,1,0,362880,299376,724140,275120,118125,29673,4410,630,45,1
%N 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.
%C A348191 A permutation p in S_n is a cube if there exists q in S_n with q^3=p.
%H A348191 Alois P. Heinz, <a href="/A348191/b348191.txt">Rows n = 0..150, flattened</a>
%e A348191 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).
%e A348191 Triangle begins:
%e A348191 [0]  1;
%e A348191 [1]  0,   1;
%e A348191 [2]  0,   1,   1;
%e A348191 [3]  0,   0,   3,    1;
%e A348191 [4]  0,   6,   3,    6,   1;
%e A348191 [5]  0,  24,  30,   15,  10,   1;
%e A348191 [6]  0,   0, 234,  105,  45,  15,  1;
%e A348191 [7]  0, 720, 504, 1134, 315, 105, 21, 1;
%p A348191 with(combinat):
%p A348191 b:= proc(n, i) option remember; expand(`if`(n=0, 1, `if`(i<1, 0,
%p A348191       add(`if`(irem(j, igcd(i, 3))<>0, 0, x^j*(i-1)!^j*
%p A348191       multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1)), j=0..n/i))))
%p A348191     end:
%p A348191 T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
%p A348191 seq(T(n), n=0..10);  # _Alois P. Heinz_, Nov 30 2021
%t A348191 multinomial[n_, k_List] := n!/Times @@ (k!);
%t A348191 b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i<1, 0,
%t A348191      Sum[If[Mod[j, GCD[i, 3]] != 0, 0, x^j*(i-1)!^j*multinomial[n,
%t A348191      Join[{n-i*j}, Table[i, {j}]]]/j!*b[n-i*j, i-1]], {j, 0, n/i}]]]];
%t A348191 T[n_] := With[{p = b[n, n]}, Table[Coefficient[p, x, i], {i, 0, n}]];
%t A348191 Table[T[n], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Dec 28 2021, after _Alois P. Heinz_ *)
%Y A348191 Columns k=0-1 give: A000007, |A194770|.
%Y A348191 Row sums give A103619.
%Y A348191 Cf. A246948.
%K A348191 nonn,tabl
%O A348191 0,9
%A A348191 _Steven Finch_, Nov 27 2021
%E A348191 More terms from _Alois P. Heinz_, Nov 30 2021