Re: Permutation of maximum cycle

Derek Holt wrote:
vincent64@xxxxxxxxx wrote:
A permutation p on (1,2,...,n) has a period (or cycle) k defined as the
minimum integer > 1 such that p^k = p. Given n, what is the largest
potential value for k?

Depends what you mean by "largest potential value".

There is no formula known for the largest order of a permutation on n
points, but it is asymptotically
exp(sqrt(n*log(n)))

This was apparently first proved by E. Landau in 1903.
The only reference I have is the German book
E. Landau, "Handbuch der Lehre von der Verteilung der Primzahlen",
Leipzig & Berlin, B.G. Teubner, 1909.

Also see sequence A000793 at the On-Line Encyclopedia of Integer
Sequences. Maybe the original poster wasn't able to find it there
because he defines the "order" of a permutation with p^k = p instead of
p^k = 1.

.