Re: problem with a necklace sequence
- From: dwasserm@xxxxxxxxxxxxx
- Date: 25 Apr 2006 16:53:46 -0700
Actually A066313 shows 3 necklaces for n = 5, and 6 necklaces for n =
6.
I believe it means that if you take any necklace, reverse its order,
and reverse the color of each bead, the result is considered equivalent
to the original necklace. This make sense physically if each bead is
red on the top and blue on the bottom, or vice versa. More likely, the
two colors are metaphors for objects that spin clockwise or
counterclockwise, or have magnetic fields in opposite directions, or
some such thing.
So the 6 necklaces with 6 beads are
a a a a a b (equivalent to a b b b b b)
a a a a b b (equivalent to a a b b b b)
a a a b a b (equivalent to a b a b b b)
a a b b a b
b b a a b a
a a a b b b
"Aperiodic" apparently means that the n-bead sequence may not consist
of a shorter sequence that repeats, e.g. a b a b a b, or a a b a a b.
This restriction makes sense physically if you think of an infinite
periodic sequence instead of a finite loop. ...ababababab... doesn't
need to be counted for n = 6 because it is counted at n = 2. In
contrast, a necklace with 6 beads ababab is physically distinguishable
from a necklace with 2 beads ab.
.
- References:
- problem with a necklace sequence
- From: lloyd
- problem with a necklace sequence
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