%I #28 Jun 22 2019 18:59:40
%S 1,1,1,3,8,38,192,1320,10176,91296,908160,9985920,119761920,
%T 1556847360,21794734080,326920043520,5230700052480,88921882828800,
%U 1600593472880640,30411275613143040,608225502973132800
%N Number of distinct classes of permutations of length n under reversal, rotation and complement to n+1.
%C Contribution from _Olivier Gérard_, Jul 31 2016: (Start)
%C Let us consider two permutations to be equivalent if they can be obtained from each other by reversal (12345->54321), cyclic rotation (12345->(23451,34512,45123,51234), n+1-complement (31254->35412), or a combination of those three transformations (they commute with each other). a(n) is the number of classes. It is strictly inferior to (n-1)! for n>1.
%C If rotation is replaced by addition of a constant modulo n, one obtains the same number of classes but not exactly the same permutations starting n=5.
%C (End)
%C Original explanation by R. Jerome : Generate all permutations of a string of length n, such as 1234, which has length 4; there are n!=24 of these. Now remove all that have cycles less than 4 characters long; if you only use cyclic notation and not array notation then of the n! possibly only (n-1)! need to be considered. Then calculate the Inverse, Vertical reflection, [VErt reflection inverse], Rotation by 180 degree and [ROt by 180 deg inverse]. If any of these already exist on the list then this permutation is not distinct. Items in []'s are unnecessary since VE(x)=V(I(x))=I(V(x))=R(x) and RO(x)=R(I(x))=I(R(x))=V(x). There are some that are rotationally symmetric and some that are vertically symmetric (only possible for even lengths), but the majority are nonsymmetric.
%H J. Gebel, <a href="/A001014/a001014.txt">Integer points on Mordell curves</a> [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]
%H Samuel Herman, Eirini Poimenidou, <a href="https://arxiv.org/abs/1905.04785">Orbits of Hamiltonian Paths and Cycles in Complete Graphs</a>, arXiv:1905.04785 [math.CO], 2019.
%H R. Jerome, <a href="https://sites.google.com/site/rayjeromejobs/permutations">Information for Unique Permutations</a>.
%e Examples of permutations (notations of R. Jerome):
%e Rotationally symmetric: x1=R(x1)=124356=VE(x1), I(x1)=165342=V(x1)=RO(x1)
%e Vertically symmetric: x2=V(x2)=132645=RO(x2)), I(x2)=154623=R(x2)=VE(x2)
%e Nonsymmetric: x3=135264, I(x3)=146253, R(x3)=152463=VE(x3), V(x3)=136425=RO(x3)
%e a(4)=3: there are 3 distinct permutations of exactly length 4, out of a field of 4!=24 possible permutations. In cyclic notation they are designated (1234), (1243) and (1324). The others, (1342), (1423) and (1432), are equal to inverses, vertical mirror images or 180-degree rotations of those 3. The remaining 18 have cycles of length 1, 2 or 3, such as (143)(2) having a permutation of length 3 and a fixed cycle and (14)(23) having 2 permutations of length 2.
%e Examples of permutation representatives (from Olivier Gerard)
%e The representative is chosen to be the first of the class in lexicographic order.
%e n=4 both cases
%e 1234,1243,1324
%e n=5 case rotation, reversal, complement
%e 12345,12354,12435,12453,12534,13425,13524,14325
%e n=5 case translation mod, reversal, complement
%e 12345,12354,12435,12453,12534,13425,13452,13524
%t (* From the formula in A099030 *)
%t a[n_] := If[n < 3, 1, 1/4 If[Mod[n, 2] == 0,((n - 1)! + (n/2 + 1) (n - 2)!!), ((n - 1)! + (n - 1)!!)]]; Table[a[n], {n, 1, 20}]
%Y Apart from initial terms, same as A099030. - _Ray Jerome_, Feb 25 2005
%Y Cf. A000939 (same idea under (rotation, addition mod n and reversal) or (rotation, addition mod n and complement)).
%Y Cf. A000940 (same idea under (rotation, addition mod n, reversal and complement)).
%Y Cf. A001710 (shifted, same idea under (rotation and reversal) or (addition mod n and complement)).
%Y Cf. A002619 (same idea under (rotation and addition mod n)).
%Y Cf. A262480 (same idea under (reversal and complement)).
%Y cf. A275527 (same idea under (rotation and complement) or (addition mod n and reversal)).
%K nonn
%O 1,4
%A _Ray Jerome_, Dec 02 2003, Jul 17 2007
%E Definition changed and cross-references added by _Olivier Gérard_, Jul 31 2016