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.

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

Offset: 0

Author

Alois P. Heinz, Aug 11 2014

Keywords

• nonn
• tabl

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

Examples

			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;
  ...
		

Links

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

Crossrefs

Columns k=0-10 give: A000007, A045531, A246189, A246190, A246191, A246192, A246193, A246194, A246195, A246196, A246197.

T(2n,n) gives A246050.

Row sums give A000312.

Main diagonal gives A000142(n-1) for n>0.

Cf. A241981, A243098.

Programs

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 *)