[r-t] irregular leadheads

[Richard Smith]

Peter King wrote:

> There are 24 (4!) possible leadheads but I shall avoid
> things like 3254 (2 lead course) or 3542 (3 lead course)
> which leaves just two families (the labelling I am using
> is out of simplicity and is not related to the usual
> nomenclature). There are the regular leadheads:
>
> a 3524
> b 5432
> c 4253

Except that b has a 2-lead course -- it is a differential
hunter.

> and the cyclic ones
> d 3452
> e 4523
> f 5234

Likewise, e has a 2-lead course.

Because the lead ends form the whole of a conjugacy class
for the
> So there are 6 ie (n-2)! exactly as you said.

As PABS points out, you are missing

  4532
  5423

Removing b and e and replacing them with these two gives the
six 4-lead lead heads.  As I said before, if you want
a non-differential (n-1)-lead course, there are (n-2)!
possible lead heads.  This is true on five bells (the six
lead heads are 3524, 4253, 3452, 5234, 4532 and 5423)
despite the fact that four is not prime.

> The question of symmetry is an interesting one I hadn't
> thought about previously but then if I restrict myself to
> certain symmetry types what does the answer become (so eg
> Woodbine has the usual symmetry but leadhead 52364 and so
> is irregular.

Let's first focus on single-hunt even bell methods with
palindromic symmetry (i.e. "normal" methods).  As per my
previous email, the number of lead ends is (n-1)(n-3)...3.
Let's assume that all of these can be obtained (this is not
always true: certainly for Little methods there is not space
to reach all of the lead ends).

We require the lead head to be an (n-1)-cycle.  This must be
reached from the lead end via a change containing exactly
two places, one of which is in first place (the treble).
There are n/2 such changes.  (This is including lead end
changes that would usually be though of as bobs, such as 14
in minor).  This gives (n/2).(n-1)(n-3)...3 lead-end /
lead-head combinations, but does not yet properly take into
account our requirement that the lead head be an
(n-1)-cycle.

Concentrating on a specific lead end change (say 12), what
propotion of possible lead ends produce a (n-1)-cycle lead
end?  Let's draw the change 12 (on six bells) as follows:

  1  2  3  4  5  6
  o  o  o--o  o--o

That is, with lines joining swapping pairs.  And let's
look at one possible lead end row, 153624, which we can draw

  1  2  3  4  5  6
            _____
           |     |
  o  o  o  o  o  o
     |________|

If we combine these two pictorial representations, we get

  1  2  3  4  5  6
            _____
           |     |
  o  o  o--o  o--o
     |________|

which is a line passing through all five working bells.
When we multiply the lead end row 153624 by the lead end
change 12, we get the 5-lead lead-head 156342.  It's easy to
show that all (n-1)-cycles (where n is even) correspond to
lines like this.

So, given the change,

  1  2  3  4  5  6
  o  o  o--o  o--o

how many ways of turning this into a 5-cycle are there?
First we need a line from 2 to one of the other (n-2)
working bells; this bell will already have a line to another
bell.  We now must join this to one of the (n-4) remaining
working bells, and so on.  This gives (n-2)(n-4)...2
different lead heads with that particular lead end change.

Allowing any lead end change, there are (n/2).(n-2)(n-4)...2
possible lead end / head combinations.  Or, if you prefer a
closed formula, 2^(n/2-1) (n/2)!.  Tabulating the first few
values:

  Bells   Le/lhs
  4            2
  6           24
  8          192
  10        1920
  12       23040
  14      322560
  16     5160960

> DoubleCambridge cyclic hasa different
> symmetry and leadhead 34562,presumably differentgroups
> fall out). Once again Ipresume the answer you gave
> previously holds if n-1 is prime.

I'll have a think about this one.  My first reaction is
that with a long enough lead, any lead head is probably
possible.

> Also is it possible to have an odd bell cyclic method with
> the usual symmetry

Only on five bells -- not on more.

> Finally there doesn't appear to be a classification
> beyond those you mentioned. So I can't find any standard
> nomenclature for cyclic type lead heads (as well as the
> other irregular ones).

As you can see from the table above, it wouldn't be feasible
to have codes for irregular methods at higher stages.  Yes,
you could have two-letter codes for Major, three-leter ones
for Royal, etc., but this doesn't strike me as particularly
useful.

When and if cyclic methods become more popular, we might
start to see the adoption of some naming convention for
cyclic-lead heads.  Until then, however, it probably isn't
that useful.

Richard