[r-t] Jump change notation

[Don Morrison]

On Sun, May 17, 2015 at 11:05 AM, Don Morrison <dfm at ringing.org> wrote:
> To summarize, for any of the four schemes there are two
> transformations we need to be able to make: (a) reading the notation
> and turning it into a permutation, and (b) taking an arbitrary
> permutation and turning it into the notation.
>
> Here's a table of what I can see regarding these choices for the four
> schemes. In it 't' indicates it is trivially easy to see how to do it;
> 'c' indicates it is relatively easy to see how to do it modulo we need
> to have some way of coming up with a canonical version of the
> notation, since there are multiple ways of notating any given
> permutation; and 'o' indicates it is open, not clear to me what an
> efficient algorithm would be.
>
> ..........(a) (b)
> scheme 1:  t   t
> scheme 2:  o   c
> scheme 3:  o   c
> scheme 4:  t   o
>
> Can anyone fill in any of the 'o's?

Sorry, I haven't been thinking clearly. Task (a) for scheme (2) is,
I think, easy, nearly the same as for ordinary changes:

Leaving aside for the moment elided leading or lying places, simply
fill in all the known positions, either because a place is being made,
or because they are included in a cycle. All the remaining positions
must occur as adjacent pairs swapping. If they don't fit, there's an
error in that chunk of extended place notation.

The amendment for elided leading or lying places is the same as for
ordinary changes:
if there are an odd number of un-filled-in positions before the first explicitly
cited position, add a leading place; if there are an odd number of
un-filled-in positions
after the last explicitly cited position add a lying place; and for
the special case of
a hunt on an even number bells at least one of the leading or lying
place must have been
explicitly cited.

Does anyone see any mistake in that?

In fact, scheme (2) has the virtue of explicitly citing every position that is
involved in something unusual: that is, either jumping, or moving without a jump
but not exchanging with the bell whose position it is taking. And it is nearly
as clear as scheme (3) in that you can immediately read the jumping bells off by
noting wherever non-consecutive bells appear adjacent to one another, counting
wrapping around as adjacency.

Have I got that right, too?




-- 
Don Morrison <dfm at ringing.org>
"A living legacy, [Bull's] propagation of the Hardanger fiddle and his
cultivation of Hardanger fiddlers helped make this traditional eight-
or nine-stringed instrument as basic to Norwegian identity as
lutefisk, and for many more appealing."
     -- David Schoenbaum, _The Violin: A Social History_