A285849 - OEIS

%I #16 Oct 18 2018 15:43:05

%S 1,0,1,0,1,1,0,2,6,1,0,6,19,18,1,0,24,100,105,40,1,0,120,508,1005,430,

%T 75,1,0,720,3528,6762,6300,1400,126,1,0,5040,24876,61572,62601,28700,

%U 3822,196,1,0,40320,219168,558548,706608,431445,105336,9114,288,1

%N Number T(n,k) of permutations of [n] with k ordered cycles such that equal-sized cycles are ordered with increasing least elements; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%C Each cycle is written with the smallest element first and equal-sized cycles are arranged in increasing order of their first elements.

%H Alois P. Heinz, <a href="/A285849/b285849.txt">Rows n = 0..140, flattened</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>

%e T(3,1) = 2: (123), (132).

%e T(3,2) = 6: (1)(23), (23)(1), (2)(13), (13)(2), (3)(12), (12)(3).

%e T(3,3) = 1: (1)(2)(3).

%e Triangle T(n,k) begins:

%e   1;

%e   0,    1;

%e   0,    1,     1;

%e   0,    2,     6,     1;

%e   0,    6,    19,    18,     1;

%e   0,   24,   100,   105,    40,     1;

%e   0,  120,   508,  1005,   430,    75,    1;

%e   0,  720,  3528,  6762,  6300,  1400,  126,   1;

%e   0, 5040, 24876, 61572, 62601, 28700, 3822, 196, 1;

%p b:= proc(n, i, p) option remember; expand(`if`(n=0 or i=1,

%p       (p+n)!/n!*x^n, add(b(n-i*j, i-1, p+j)*(i-1)!^j*combinat

%p       [multinomial](n, n-i*j, i$j)/j!^2*x^j, j=0..n/i)))

%p     end:

%p T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2, 0)):

%p seq(T(n), n=0..12);

%t multinomial[n_, k_List] := n!/Times @@ (k!);

%t b[n_, i_, p_] := b[n, i, p] = Expand[If[n == 0 || i == 1, (p + n)!/n!*x^n, Sum[b[n - i*j, i - 1, p + j]*(i - 1)!^j*multinomial[n, Join[{n - i*j}, Table[i, j]]]/j!^2*x^j, {j, 0, n/i}]]];

%t T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, n, 0]];

%t Table[T[n], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Apr 28 2018, after _Alois P. Heinz_ *)

%Y Columns k=0-10 give: A000007, A104150, A285853, A285854, A285855, A285856, A285857, A285858, A285859, A285860, A285861.

%Y Row sums give A196301.

%Y Main diagonal and first lower diagonal give: A000012, A002411.

%Y T(2n,n) gives A285862.

%Y Cf. A132393, A285824.

%K nonn,tabl

%O 0,8

%A _Alois P. Heinz_, Apr 27 2017