[r-t] Irregular leadhead orders

[Phillip Barnes]

Very interesting

However - in point 3 I see the phrase “… , and what lead end orders are allowable”. I thought we were aiming for (simple and) permissive. Isn’t allowable a bit regressive in that regard?

Obviously I may have missed the point :)

Phil


On 10 June 2018 at 12:22:48, Mark Davies (mark at snowtiger.net) wrote:

The Plain Bob leadhead order with its associated "natural coursing  
order" is all-pervasive in the world of method-ringing. It has the  
following interesting properties:

1. It can be produced by methods which are symmetric about the half-lead.

2. It can be produced by methods with either 12 or 1n places at the lead  
end.

3. It extends naturally to all stages.

On n bells there are (n-3)! possible leadhead orders for  
non-Differential treble-dominated methods. I think the PB order is the  
only one which possesses all three properties above, but there are  
others which share two out of three properties. For instance the  
"cyclic" order, characterized by leadheads such as 134562, 13456782,  
1345678902 etc, can be produced either by 12 or 1n, and extends  
naturally to all stages. However it cannot be produced (in the normal  
way) by a symmetric method.

There are plenty of other leadhead orders which can be produced by  
symmetric methods and which allow one PN at the lead end, e.g. 12 only.  
The following Minor and Major orders fall into this category:

65423: 142563, 154632, 165324, 136245
8746253: 16572483, 14286735, 17634852, 18457326, 13728564, 15863247
8745263: 15674283, 14287536, 17538462, 18463725, 13726854, 16852347
8736254: 16752384, 13826745, 17463852, 18537426, 14278563, 15684237
8735264: 15763284, 13827546, 17458362, 18634725, 14276853, 16582437
8764235: 14263785, 16472853, 17684532, 18756324, 15837246, 13528467
8763245: 13624785, 16732854, 17863542, 18576423, 15487236, 14258367
8754236: 14257386, 15478263, 17586432, 18763524, 16832745, 13624857

Here I list the coursing order followed by the non-rounds leadheads of  
the group. The coursing order is taken by examining the bells falling  
into positions (n)(n-1) at the leadheads, normalised to put the tenors  
first.

My questions are:

1. Is there a relationship between the 6-bell order and one or more of  
the Major orders, in the same way that the PB and cyclic orders  
naturally "extend" to higher numbers? If so, what would the Royal,  
Maximus etc extensions look like?

2. Is there a general way of finding such a relationship between  
irregular leadhead orders at different stages?

3. Should any such relationship naturally respect the "structural"  
properties such as whether a symmetric method is possible, and what lead  
ends PNs are allowable?

Is there any existing group theory which can help here? All these  
leadhead orders are effectively cyclic subgroups of Sn. Is there any  
characterization of such subgroups which can help?

MBD


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