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A285849 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.

%I A285849 #16 Oct 18 2018 15:43:05
%S A285849 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 A285849 75,1,0,720,3528,6762,6300,1400,126,1,0,5040,24876,61572,62601,28700,
%U A285849 3822,196,1,0,40320,219168,558548,706608,431445,105336,9114,288,1
%N A285849 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 A285849 Each cycle is written with the smallest element first and equal-sized cycles are arranged in increasing order of their first elements.
%H A285849 Alois P. Heinz, <a href="/A285849/b285849.txt">Rows n = 0..140, flattened</a>
%H A285849 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%e A285849 T(3,1) = 2: (123), (132).
%e A285849 T(3,2) = 6: (1)(23), (23)(1), (2)(13), (13)(2), (3)(12), (12)(3).
%e A285849 T(3,3) = 1: (1)(2)(3).
%e A285849 Triangle T(n,k) begins:
%e A285849   1;
%e A285849   0,    1;
%e A285849   0,    1,     1;
%e A285849   0,    2,     6,     1;
%e A285849   0,    6,    19,    18,     1;
%e A285849   0,   24,   100,   105,    40,     1;
%e A285849   0,  120,   508,  1005,   430,    75,    1;
%e A285849   0,  720,  3528,  6762,  6300,  1400,  126,   1;
%e A285849   0, 5040, 24876, 61572, 62601, 28700, 3822, 196, 1;
%p A285849 b:= proc(n, i, p) option remember; expand(`if`(n=0 or i=1,
%p A285849       (p+n)!/n!*x^n, add(b(n-i*j, i-1, p+j)*(i-1)!^j*combinat
%p A285849       [multinomial](n, n-i*j, i$j)/j!^2*x^j, j=0..n/i)))
%p A285849     end:
%p A285849 T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2, 0)):
%p A285849 seq(T(n), n=0..12);
%t A285849 multinomial[n_, k_List] := n!/Times @@ (k!);
%t A285849 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 A285849 T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, n, 0]];
%t A285849 Table[T[n], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Apr 28 2018, after _Alois P. Heinz_ *)
%Y A285849 Columns k=0-10 give: A000007, A104150, A285853, A285854, A285855, A285856, A285857, A285858, A285859, A285860, A285861.
%Y A285849 Row sums give A196301.
%Y A285849 Main diagonal and first lower diagonal give: A000012, A002411.
%Y A285849 T(2n,n) gives A285862.
%Y A285849 Cf. A132393, A285824.
%K A285849 nonn,tabl
%O A285849 0,8
%A A285849 _Alois P. Heinz_, Apr 27 2017