[r-t] irregular leadheads
King, Peter R
peter.king at imperial.ac.uk
Fri Dec 3 15:36:22 UTC 2004
I too was excluding differential lead heads. The number of
non-differential (n-1)-lead lead heads is (n-2)!
irrespective of whether (n-1) is prime.
You right of course. I have now had time to think about this instead of dashing around between meetings. I alsonow realise what question I should have asked which, I think has a different answer.
I'll use doubles as an example because there you can write everything down.
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
and the cyclic ones
d 3452
e 4523
f 5234
So there are 6 ie (n-2)! exactly as you said. So this is the answer to the question as I stated it and applies regardless of n.
However whilst I can have an a type method (plain bob) or a c type method (St Simons) I can't have a b type method (double bob would be this) given the restriction I had of wanted a 4 (n-1) lead course. Similarly I can have d or f but not e. So yes there are 6 lead heads (answer to question as asked) but there are only 4 method types (ignoring 2nds/ nths (or 1sts for doubles) variants). So the question I should have asked is how many lead head types (that satisfy the criterion of giving an n-1 lead course) are there for n bells. Presumably this is just (n-2)! for n-1 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.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.
Also is it possible to have an odd bell cyclic method with the usual symmetry (eg on 5 5.1.5.3.345 lh1 which is St Simons with 345 made at the 1/2 lead gives lh 3452, it'snot very elegant but I just wrote it down now). Just an aside and probably like the other rushed thoughts incorrect!
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).
I hope this has clarified the misunderstanding
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