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A242027
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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.
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14
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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
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OFFSET
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0,5
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LINKS
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EXAMPLE
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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;
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MAPLE
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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);
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MATHEMATICA
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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 *)
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CROSSREFS
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Columns k=0-10 give: A000007, A241980 for n>0, A246283, A246284, A246285, A246286, A246287, A246288, A246289, A246290, A246291.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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