[r-t] ringing related paper

[Alexander Holroyd]

On Sun, 23 Aug 2015, Philip Earis wrote:

> Prof. Holroyd:
> "Here is a mathematics paper related to change ringing theory that people 
> might be amused to see.  The main result can be interpreted as: any 
> permutation may be achieved by a link method with a bounded number of changes 
> of direction for each bell, regardless of the stage (specifically, 10 changes 
> of direction per bell is enough)..."
>
> Interesting.  What is the maximum number of changes of direction a bell needs 
> at stages 8, 10, 12?  What permutations at these stages require the maximum 
> changes of direction?

Good questions.  I don't know the answer to these.  They could be answered 
by exhaustive search.  Furthermore we don't know whether 8 or 6 changes of 
direction per bell suffice at every stage (10 is just the upper bound we 
can prove).  There are permutations that require 6 changes of direction 
for some bell.

A related fact is that every permutation on 6 or fewer bells can be 
achieved with at most 4 changes of direction per bell - indeed, each bell 
only needs to hunt up at most once and down at most once.  We have a 
precise characterization of all permutations that can be achieved by such 
a method at every stage (but it is quite complicated).  See ourother 
paper, reference [4].