OFFSET
0,5
COMMENTS
T(0,0) = 1 by convention.
In general, number of endofunctions on [n] where the largest cycle length equals k is asymptotic to (k*exp(H(k)) - (k-1)*exp(H(k-1))) * n^(n-1), where H(k) is the harmonic number A001008/A002805, k>=1. The multiplicative constant is (for big k) asymptotic to 2*k*exp(gamma), where gamma is the Euler-Mascheroni constant (see A001620 and A073004). - Vaclav Kotesovec, Aug 21 2014
LINKS
Alois P. Heinz, Rows n = 0..140, flattened
EXAMPLE
Triangle T(n,k) begins:
1;
0, 1;
0, 3, 1;
0, 16, 9, 2;
0, 125, 93, 32, 6;
0, 1296, 1155, 500, 150, 24;
0, 16807, 17025, 8600, 3240, 864, 120;
0, 262144, 292383, 165690, 72030, 24696, 5880, 720;
...
MAPLE
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 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, min(j, k)), j=0..n):
T:= (n, k)-> A(n, k) -`if`(k=0, 0, A(n, k-1)):
seq(seq(T(n, k), k=0..n), n=0..10);
MATHEMATICA
multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 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, Min[j, k]], {j, 0, n}]; T[0, 0] = 1; T[n_, k_] := A[n, k] - If[k == 0, 0, A[n, k-1]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 06 2015, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Aug 10 2014
STATUS
approved