[r-t] Bobs only Erin Triples (was: Composition search)

[Richard Smith]

>> On Sat, Nov 28, 2015 at 2:53 AM Richard Smith wrote:
>>> If a multi-part bobs-only extent of Erin exists, then 
>>> the number of parts must be one of 168, 60, 24, 21, 20, 
>>> 12, 10, 8, 7, 6, 5, 4, 3, 2 or 1.
>>
>> I'm having trouble getting my head around what the part 
>> heads of a six part, bobs only extent of Erin might be; 
>> the only ones I can imagine would have a 3-cycle in them. 
>> What am I missing?
>
> 1234567
> 2143756
> 1234675
> 2143567
> 1234756
> 2143675
> 1234567

No.  That doesn't work as it contains a single 3-cycle, c.f. 
1234675.

The group in question has all permutations of 123 while 456 
permute in phase:

   123456  213546  231564  321654  312645  132465

Although half of the elements are of order 3 and contain 
3-cycles, none are just a 3-cycle and so they're not 
immediately rule out by the falseness.

> A regular 12-part is  a bit harder...

It can be done in two ways, and both need searching 
separately.

In one, the pairs 12, 34 and 56 rotate, and swap to keep 
parity:

   123456  345612  561234
   124365  346521  562143
   213465  435621  651243
   214356  436512  652134

The other has all even permutations of 1234 (which would 
include 3-cycle), together with a linked 3-cycle on 567:

   1234567  1423756  3241756
   3412567  2314756  4132756
   1342675  2431675  4321567
   4213675  3124675  2143567

As always, Brian Price's 'Composition of Peals in Parts' 
provides the definitive guide to these things.

RAS