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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. 14
1, 0, 1, 0, 4, 0, 24, 3, 0, 206, 50, 0, 2300, 825, 0, 31742, 14794, 120, 0, 522466, 294987, 6090, 0, 9996478, 6547946, 232792, 0, 218088504, 160994565, 8337420, 0, 5344652492, 4355845868, 299350440, 151200, 0, 145386399554, 128831993037, 11074483860, 18794160 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

LINKS

Alois P. Heinz, Rows n = 0..140, flattened

EXAMPLE

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

Triangle T(n,k) begins:

00 :  1;

01 :  0,          1;

02 :  0,          4;

03 :  0,         24,          3;

04 :  0,        206,         50;

05 :  0,       2300,        825;

06 :  0,      31742,      14794,       120;

07 :  0,     522466,     294987,      6090;

08 :  0,    9996478,    6547946,    232792;

09 :  0,  218088504,  160994565,   8337420;

10 :  0, 5344652492, 4355845868, 299350440, 151200;

MAPLE

with(combinat):

b:= proc(n, i, k) option remember; `if`(n=0, `if`(k=0, 1, 0),

      `if`(i<1 or k<1, 0, add((i-1)!^j*multinomial(n, n-i*j, i$j)/j!*

      b(n-i*j, i-1, k-`if`(j=0, 0, 1)), j=0..n/i)))

    end:

T:= (n, k)-> add(binomial(n-1, j-1)*n^(n-j)*b(j$2, k), j=0..n):

seq(seq(T(n, k), k=0..floor((sqrt(1+8*n)-1)/2)), n=0..14);

MATHEMATICA

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 *)

CROSSREFS

Columns k=0-10 give: A000007, A241980 for n>0, A246283, A246284, A246285, A246286, A246287, A246288, A246289, A246290, A246291.

Row sums give A000312.

T(A000217(n),n) gives A246292.

Cf. A003056, A060281, A218868 (the same for permutations).

Sequence in context: A229827 A295839 A243270 * A329891 A057402 A269214

Adjacent sequences:  A242024 A242025 A242026 * A242028 A242029 A242030

KEYWORD

nonn,tabf

AUTHOR

Alois P. Heinz, Aug 11 2014

STATUS

approved

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Last modified May 30 08:43 EDT 2020. Contains 334712 sequences. (Running on oeis4.)