[r-t] Fwd: bobs-only Grandsire Triples

[Andrew Johnson]

> From: Roy Dyckhoff <roy.dyckhoff at googlemail.com>
> 
> It would be interesting if you could share the paper mentioned with the 
list?
> Now done.
>  
> See https://dl.dropboxusercontent.com/u/9941616/Bobs-onlyGT.pdf
> 
> RD
The paper says:

>Also, there remains the challenge, mentioned in a footnote, of showing 
that every row of Grandsire Cinques
>can be reached by an appropriate sequence of calls (i.e. the permutations 
P and B generate the group of rows).
>There is a result in group theory of which this is a consequence; the 
challenge is to find a simple argument,
>not based on group theory or an impossible exhaustive case analysis. Such 
an argument can be expected to
>apply also to Grandsire Triples.

It's not an impossible exhaustive case analysis - generating the group is 
pretty straightforward
and entirely possible because the result isn't that big. The treble is 
fixed (10 working bells),
and P and B are even, so the biggest it could be is A10 of size 1814400.

With my simple program for generating groups it only took a couple of 
minutes to show this.

rexx gengroup.cmd 12537496E80 175293E4068 >g11
$ wc g11
 1814400  1814400 23587200 g11

You then have to show that every row (not just the treble lead heads) can 
be reached, but that
would also be simple to do exhaustively, and you can also show that each 
row can only be obtained by
one in-course lead head.

Even generating A11 or S11 e.g for Stedman Cinques is feasible with a 
computer.

The second lead of a plain course of Grandsire Cinques (PP) is
127593E4068
e.g. a 9-cycle.

compare that to a bob lead B
175293E4068

which similar apart from a 3-cycle which might help in forming a 
hand-waving explanation.

Compare with generating Sn by a cyclic permutation of order n, and 
swapping of two bells, 
and An for odd n by a cyclic permutation of order n and a 3-way cycle, and 
for
even n by an generator formed from a cyclic permutation of order n applied 
twice 
(to get an even permutation) and a 3-way cycle.

I think the above also applies with a cyclic permutation of order n-1, 
where the swapping
includes the unaffected bell. E.g.:
Generate Sn by a cyclic permutation of order n-1, and swapping of two 
bells including the
unaffected bell, 
and An for even n by a cyclic permutation of order n-1 and a 3-way cycle 
involving the
unaffected bell, and for odd n by an generator formed from a cyclic 
permutation of 
order n-1 applied twice (to get an even permutation) and a 3-way cycle 
involving the
unaffected bell.

Andrew Johnson




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