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Permutations of maximum period Problem: Let p be a permutation on [n] = (1,...,n). We are considering operations on the permutation group of order n. Let us define p2 as p o p, and more generally pk+1 as p o pk. For instance, if n=3 and p = (3,1,2), then p2 = (2,3,1) is obtained by applying p twice, and p3 = (1,2,3) = I. We shall use the letter I to denote the identity permutation: I = (1,2,...,n). Then the period is the minimum integer k > 0 such that pk = I. For a given n, we are interested in finding permutations with maximum period. Let us note that the permutation (2,3,4,...,n-1,n,1) always has a period of n. As a consequence, the following permutation with n=12 with 3 cycles (sub permutations) defined as p=(2,3,1|5,6,7,4|9,10,11,12,8) has a period 3 x 4 x 5 = 60. This is true because the three sub-periods - 3, 4 and 5 - are mutually prime. In that case the period is the product of the sub-periods. More generally, if we had m cycles, the period could be as large as (n/m)m. This expression attains its maximum when m = 0.368 x n (0.368 = 1/e). In other words, the maximum is achieved when we have about m/3 cycles with mutually prime periods, regardless of n. The maximum period is then of the order en/e < n! Is my reasoning correct? I am trying to create random permutations to encrypt a string s, in the context of fraud detection. The idea is that permutations with large periods, if well chosen, will be very difficult to identify. Let's say that p has an even period v. Then if you only know pv/2 but not p nor v, it is almost impossible to identify p and v. Once a hacker knows p and v, he can easly retrieve the original string s. Contributions: From David Bernier: This is related to Landau's function g(n) , n being the number of elements in the set getting permuted. From Derek Holt: I have a C program to identify permutations of maximum order. For example, for n=10000, the maximum 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