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A242027 Number T(n,k) of endofunctions on [n] with cycles of k distinct lengths; triangle T(n,k), n>=0, 0<=k<=A003056(n), read by rows.

%I A242027 #20 Feb 18 2017 10:56:01
%S A242027 1,0,1,0,4,0,24,3,0,206,50,0,2300,825,0,31742,14794,120,0,522466,
%T A242027 294987,6090,0,9996478,6547946,232792,0,218088504,160994565,8337420,0,
%U A242027 5344652492,4355845868,299350440,151200,0,145386399554,128831993037,11074483860,18794160
%N A242027 Number T(n,k) of endofunctions on [n] with cycles of k distinct lengths; triangle T(n,k), n>=0, 0<=k<=A003056(n), read by rows.
%H A242027 Alois P. Heinz, <a href="/A242027/b242027.txt">Rows n = 0..140, flattened</a>
%e A242027 T(3,2) = 3: (1,3,2), (3,2,1), (2,1,3).
%e A242027 Triangle T(n,k) begins:
%e A242027 00 :  1;
%e A242027 01 :  0,          1;
%e A242027 02 :  0,          4;
%e A242027 03 :  0,         24,          3;
%e A242027 04 :  0,        206,         50;
%e A242027 05 :  0,       2300,        825;
%e A242027 06 :  0,      31742,      14794,       120;
%e A242027 07 :  0,     522466,     294987,      6090;
%e A242027 08 :  0,    9996478,    6547946,    232792;
%e A242027 09 :  0,  218088504,  160994565,   8337420;
%e A242027 10 :  0, 5344652492, 4355845868, 299350440, 151200;
%p A242027 with(combinat):
%p A242027 b:= proc(n, i, k) option remember; `if`(n=0, `if`(k=0, 1, 0),
%p A242027       `if`(i<1 or k<1, 0, add((i-1)!^j*multinomial(n, n-i*j, i$j)/j!*
%p A242027       b(n-i*j, i-1, k-`if`(j=0, 0, 1)), j=0..n/i)))
%p A242027     end:
%p A242027 T:= (n, k)-> add(binomial(n-1, j-1)*n^(n-j)*b(j$2, k), j=0..n):
%p A242027 seq(seq(T(n, k), k=0..floor((sqrt(1+8*n)-1)/2)), n=0..14);
%t A242027 multinomial[n_, k_] := n!/Times @@ (k!); b[n_, i_, k_] := b[n, i, k] = If[n == 0, If[k==0, 1, 0], If[i<1 || k<1, 0, Sum[(i-1)!^j*multinomial[n, Join[ {n-i*j}, Array[i&, j]]]/j!*b[n-i*j, i-1, k-If[j==0, 0, 1]], {j, 0, n/i}]] ]; T[0, 0] = 1; T[n_, k_] := Sum[Binomial[n-1, j-1]*n^(n-j)*b[j, j, k], {j, 0, n}]; Table[T[n, k], {n, 0, 14}, {k, 0, Floor[(Sqrt[1+8n]-1)/2]}] // Flatten (* _Jean-François Alcover_, Feb 18 2017, translated from Maple *)
%Y A242027 Columns k=0-10 give: A000007, A241980 for n>0, A246283, A246284, A246285, A246286, A246287, A246288, A246289, A246290, A246291.
%Y A242027 Row sums give A000312.
%Y A242027 T(A000217(n),n) gives A246292.
%Y A242027 Cf. A003056, A060281, A218868 (the same for permutations).
%K A242027 nonn,tabf
%O A242027 0,5
%A A242027 _Alois P. Heinz_, Aug 11 2014