%I A089066
%S A089066 1,1,1,3,8,38,192,1320,10176,91296,908160,9985920,119761920,1556847360,
%T A089066 21794734080,326920043520,5230700052480,88921882828800,1600593472880640,
%U A089066 30411275613143040,608225502973132800
%N A089066 Number of unique permutations of length n.
%C A089066 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 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 
               unique. 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 A089066 R. Jerome, <a href="http://Ray.Jerome.Jobs.googlepages.com/permutations">
               Information for Unique Permutations</a>.
%e A089066 Examples of permutations:
%e A089066 Rotationally symmetric: x1=R(x1)=124356=VE(x1), I(x1)=165342=V(x1)=RO(x1)
%e A089066 Vertically symmetric: x2=V(x2)=132645=RO(x2)), I(x2)=154623=R(x2)=VE(x2)
%e A089066 Nonsymmetric: x3=135264, I(x3)=146253, R(x3)=152463=VE(x3), V(x3)=136425=RO(x3)
%e A089066 a(4)=3: there are 3 unique 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.
%Y A089066 Apart from initial terms, same as A099030. - Raymond L. Jerome (raymondjerome(AT)hotmail.com), 
               Feb 25 2005
%Y A089066 Sequence in context: A147523 A123981 A123985 this_sequence A099030 A106558 
               A065914
%Y A089066 Adjacent sequences: A089063 A089064 A089065 this_sequence A089067 A089068 
               A089069
%K A089066 nonn
%O A089066 1,4
%A A089066 Raymond L. Jerome (jeromer(AT)tycoelectronics.com), Dec 02 2003, Jul 
               17 2007