[r-t] irregular leadheads

[Richard Smith]

Peter King:
> How many leadheads are there on n bells which (keep the
> treble fixed) and lead to courses of n-1 leads

Me:
> [...] So there's the answer: (n-2)!.

Peter King:
> That's what I thought and is true for n-1 prime but for Royal (for
> example) there are only 6 regular lheads (and presumably each family of
> irregular ones has only 6 members) to avoid the cycles of 3. Or am I
> wrong?

Your right that there are only 6 regular lead heads for
Royal, but this does not mean the answer is not still
(n-2)!.  Try writing down the place bell order for one of
the short-course regular Royal lead heads.  It will be
something like

  2 -> 8 -> 7 -> 2,
  3 -> 6 -> 9 -> 3,
  4 -> 0 -> 5 -> 4.

That is, it is (obviously) not a single list sequence.  Any
9-cycle can be written as a single sequence of place bells
starting say with the tenor.  Tenths place bell can become
any of the 9 other place bells, which can become any of 8
remaining place bells, until only one remains.  That place
bell becomes 10ths place bell.  So the number of 9-cycle
lead heads is still (n-2)! even though there are few regular
lead heads than might be expected.

Richard