A246220 Number of endofunctions on [n] where the largest cycle length equals 10.

362880, 43908480, 3448811520, 228012744960, 13954338478080, 827512686000000, 48753634065776640, 2895879112057451520, 174984885490926551040, 10817178515493080290560, 686533182382689959116800, 44833266187415969387604480, 3016487768851293040555130880

Offset: 10

Author

Alois P. Heinz, Aug 19 2014

Keywords

• nonn

Comments

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. - Vaclav Kotesovec, Aug 21 2014

Links

• Alois P. Heinz, Table of n, a(n) for n = 10..200

Crossrefs

Column k=10 of A241981.

Programs

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):
a:= n-> A(n, 10) -A(n, 9):
seq(a(n), n=10..25);

Formula

a(n) ~ (10*exp(7381/2520) - 9*exp(7129/2520)) * n^(n-1). - Vaclav Kotesovec, Aug 21 2014