Re: Permutation of maximum cycle


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".

Please reply directly to vincentg at
datashaping.com. Thank you.

You should read the Newsgroup if you expect to get answers! (You could
also try searching the Newsgroup - this question has been asked many
times over the years!)

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.

I have a simple-minded C program, which will calculate this for n up
to about 25000.

For example, for n=10000, the maximal order is

317*313*311*307*293*283*281*277*271*269*263*257*251*241*239*233*229*227*
223*211*199*197*193*191*181*179*173*167*163*157*151*149*139*137*131*127*
113*109*107*103*101*97*89*83*79*73*71*67*61*59*53*47*43*41*37*31*29*23*
19*17*169*121*49*25*81*64 =

837248314781153836222627527908638519798608214039954646068895157070382622
91218453286312511300740953193197240510692638318426636506681182400

Derek Holt.

.

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