1,4

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.

R. Jerome, Information for Unique Permutations.

Examples of permutations:

Rotationally symmetric: x1=R(x1)=124356=VE(x1), I(x1)=165342=V(x1)=RO(x1)

Vertically symmetric: x2=V(x2)=132645=RO(x2)), I(x2)=154623=R(x2)=VE(x2)

Nonsymmetric: x3=135264, I(x3)=146253, R(x3)=152463=VE(x3), V(x3)=136425=RO(x3)

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.

Apart from initial terms, same as A099030. - Raymond L. Jerome (raymondjerome(AT)hotmail.com), Feb 25 2005

Sequence in context: A147523 A123981 A123985 this_sequence A099030 A106558 A065914

Adjacent sequences: A089063 A089064 A089065 this_sequence A089067 A089068 A089069

nonn

Raymond L. Jerome (jeromer(AT)tycoelectronics.com), Dec 02 2003, Jul 17 2007