[r-t] ringing related paper
Alexander Holroyd
holroyd at math.ubc.ca
Fri Sep 4 20:29:35 UTC 2015
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].
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