[r-t] Grandsire Minor
Robert Bennett
rbennett at woosh.co.nz
Mon Feb 18 22:55:38 UTC 2013
Robert Bennett wrote:
>
> I believed that Rankine's theorem (from my MIS-reading of his 1948
paper)
> applies only to odd numbers of bells.
No, it's nothing to do with number of bells. Rankin's theorem applies
to
abstract groups without reference to the order of the symmetric group
that
they are subgroups of. It states that in the directed Cayley graph
generated by a and b, if a^-1 b has odd order then in any partition of
the
vertices into disjoint oriented cycles, the number of cycles has the
same
parity as the index of the cyclic subgroup generated by a.
This is a very clear explanation. Thanks.
As I said, this does indeed apply to grandsire triples, showing that
no
partition into an odd number of bobs-only blocks is possible. This is
because (plain)^-1 bob has order 5, i.e. Q-sets have 5 elements. In
grandsire minor, Q-sets have 4 elements, which is why it does not
apply
here.
Even better! An explanation for non-mathematicians.
> Grandsire minor the Q-sets have 4 members and the courses added on
> 3 at a time (and 1 at a time should also be possible?).
>
If one round block of Grandsire Minor contains the first and third
member of a Q set and the another block contains the 2nd and 4th
member of the same Q set, then by bobbing those Q set members, the two
blocks are combined into one. This should allow a few extra
possibilities, e.g. a 1440 should be possible.
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