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

Richard Smith richard at ex-parrot.com
Sun Nov 29 12:03:02 UTC 2015

```>> 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 isĀ  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
```