[r-t] Jump change notation

[Don Morrison]

On Sun, May 17, 2015 at 2:47 AM, Glenn Taylor
<gaataylor at blueyonder.co.uk> wrote:
> FWIW, when Cambridge Jump was first published in the RW in about 1979 the
> notation used was along the lines of:
>
> -3-(4.2)-2 etc.
>
> the "novelty" being that the bracketed part indicated that the next row was
> the end result of applying 14 followed by 12. Whilst less mathematical and
> perhaps a little more cumbersome than the notation used for permutations and
> cycles, it does have the benefit of retaining current notation (apart from
> the need to understand the purpose of the brackets) and thereby being
> "accessible" to the non-mathematical interested party.

Thank you. I'd completely forgotten about this.

It has the virtue of being entirely clear how to go from the notation
to a permutation. On the other hand, it seems exceedingly unclear how
to go from a permutation to the notation! For the simple treble jump
cases it works easily, but for more arbitrary jump changes you end up
having to compose more than two ordinary changes. In general do you
not have to assemble a sort of mini-composition or mini-method to get
between two arbitrary rows? And in general do you not have a host of
alternative possibilities to choose between (infinite, I think, though
you could undoubtedly come up with some rules that at least bound
things, probably to no more than N-1 ordinary changes to be composed,
where N is the stage)? Is there an obvious choice for a canonical
representative among them?

I'm intrigued by your implication that it is more accessible than the
alternatives. Ander's version simply names the bells (well, places)
that jump. That seems perfectly accessible to the naive user. Is it
that it is unclear how to figure out what the other bells do that
makes it seem more "mathematical" to you? And the brute force of
simply writing down the permutation seems well within the normal scope
of what ringers do--it's the same as dealing with a lead head, course
head or coursing order--what am I missing that makes that less
accessible?

Anyway, as part of seriously considering this fourth possibility, does
anyone have a suitable algorithm for going from an arbitrary
permutation to something in this notation? The opposite need as in
schemes (2) and (3) in my previous message.

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?



-- 
Don Morrison <dfm at ringing.org>
"Music is part of being human, and there is no human culture
in which it is not highly developed and esteemed."
                                  -- Oliver Sacks, _Musicophilia_