A246197 Number of endofunctions on [n] where the smallest cycle length equals 10.

362880, 39916800, 2874009600, 175394419200, 9967384627200, 551675124000000, 30471021291110400, 1703458301210265600, 97213825272736972800, 5693251850259515942400, 343266609438110040883200, 21349182724237331665228800, 1371132989012608561874534400

Offset: 10

Author

Alois P. Heinz, Aug 18 2014

Keywords

• nonn

Comments

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

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

Crossrefs

Column k=10 of A246049.

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):
a:= n-> A(n, 10) -A(n, 11):
seq(a(n), n=10..25);

Formula

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