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A246049 Number T(n,k) of endofunctions on [n] where the smallest cycle length equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 14
1, 0, 1, 0, 3, 1, 0, 19, 6, 2, 0, 175, 51, 24, 6, 0, 2101, 580, 300, 120, 24, 0, 31031, 8265, 4360, 2160, 720, 120, 0, 543607, 141246, 74130, 41160, 17640, 5040, 720, 0, 11012415, 2810437, 1456224, 861420, 430080, 161280, 40320, 5040 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
T(0,0) = 1 by convention.
In general, number of endofunctions on [n] where the smallest cycle length equals k is asymptotic to (exp(-H(k-1)) - exp(-H(k))) * n^n, where H(k) is the harmonic number A001008/A002805, k>=1. - Vaclav Kotesovec, Aug 21 2014
LINKS
EXAMPLE
Triangle T(n,k) begins:
1;
0, 1;
0, 3, 1;
0, 19, 6, 2;
0, 175, 51, 24, 6;
0, 2101, 580, 300, 120, 24;
0, 31031, 8265, 4360, 2160, 720, 120;
0, 543607, 141246, 74130, 41160, 17640, 5040, 720;
...
MAPLE
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i>n, 0,
add((i-1)!^j*multinomial(n, n-i*j, i$j)/j!*
b(n-i*j, i+1), j=0..n/i)))
end:
A:= (n, k)-> add(binomial(n-1, j-1)*n^(n-j)*b(j, k), j=0..n):
T:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), A(n, k) -A(n, k+1)):
seq(seq(T(n, k), k=0..n), n=0..12);
MATHEMATICA
multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i>n, 0,
Sum[(i-1)!^j*multinomial[n, {n-i*j, Sequence @@ Table[i, {j}]}]/j!*
b[n-i*j, i+1], {j, 0, n/i}]]];
A[n_, k_] := Sum[Binomial[n-1, j-1]*n^(n-j)*b[j, k], {j, 0, n}];
T[n_, k_] := If[k == 0, If[n == 0, 1, 0], A[n, k] - A[n, k+1]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jan 06 2015, after Alois P. Heinz *)
CROSSREFS
T(2n,n) gives A246050.
Row sums give A000312.
Main diagonal gives A000142(n-1) for n>0.
Sequence in context: A241981 A147723 A110518 * A316773 A006837 A158782
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Aug 11 2014
STATUS
approved

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Last modified May 24 13:35 EDT 2024. Contains 372773 sequences. (Running on oeis4.)