[r-t] Irregular leadhead orders

[Mark Davies]

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