[r-t] ringing the Mathieu group
Alexander Holroyd
holroyd at math.ubc.ca
Sun May 22 20:43:09 UTC 2011
Hi Colin,
Thanks for these interesting suggestions! This was just the sort of thing
I was hoping to provoke...
Regarding, e.g. suggestion 1 for the caters extent: what makes you so
confident this can be done? (I think I share this confidence, but I'm not
sure how to justify it). Certainly there are "plenty of Q-sets" (only a
few are missing, arising from the "magic block"), and there is no
(obvious) parity barrier. But why does this imply that it is possible to
join everything to one block? A general argument (rigorous or heuristic)
would be much more interesting than just checking it by giving an example
in this particular case. Ditto for the second caters suggestion...
I think the call you mention for the Mathieu group (125T in place of 18ET)
actually gives Q-sets linking 4 courses, not 8 (alternating these two pns
amounts to two lots of plain hunt on 4). In any case, no parity barrier,
as you say. Again, the question is: how can we tell whether it is
possible, other than a very tedious verification?
In both cases, another natural approach is the usual subgroup-based
"composition of peals in parts" method. Given the large size of the
problem, it mighjt be helpful to do this in several "levels", starting
with a subgroup of a subgroup. This might even yield methods that someone
would consider ringing(!) Any ideas for a specific plan in this
direction, anyone?
Ander
On Sat, 30 Apr 2011, Wyld Family e-mail wrote:
> I think the following strategies could work. If they do not it may take some
> tedious arithmetic to find out.
>
> I have two suggestions for the caters problem. Both are based on organising
> the required changes in to courses of original and linking them together
> using 5ths place calls (bobs). In the first I suggest joining, alternately,
> forward and backward hunting courses with a 5ths place i.e.
> 9.1.9.1.9.1.9.1.9.1.9.1.9.1.9.1.9.5.1.9.1.9.1.9.1.9.1.9.1.9.1.9.1.9.1.5
> repeated 8 times. This joins together 18 courses leaving an even number to
> be introduced by Q sets of 5ths place bobs. There are 18 Q sets that can not
> be bobbed having been used in the basic block but that leaves at least 8
> routes into the courses affected.
>
> The second strategy is simply to use the 5th place bobs instead of either 9
> or 1 with forward hunting throughout. The usual constraints on Q sets of
> bobs apply i.e. that the number of blocks into which the changes are
> organised is changed by an even number. Since the extent requires 10080
> courses it might at first glance seem that you would inevitably end up with
> the extent in two parts. There are however 20160 courses to choose from
> since each course can be rung in both directions. There is no obvious reason
> why adhering to the Q set rule you should not be able to reduce this larger
> collection to two parts each containing the required changes once. The two
> parts would be mirror images of each other and some of the courses would be
> rung partly in one direction and partly in the other. Since this is how 720s
> of treble bob minor and 5040s of plain bob triples work it seems worth trying
> here.
>
> The Mathieu group may be easier to handle. Again I suggest using x and one
> of the other place notations alternately to organise the changes into
> courses. Use the other place notation as a call leaving the xs unaffected.
> If I have worked it out correctly a Q set of such calls links in 7 courses
> i.e. an odd number so it may be possible simply to join together the required
> courses. If not the option to use parts of some courses rung backwards
> leading, as with the caters, to two mirror image blocks may overcome the
> problem.
>
> More interesting, at least from a ringing point of view, might be to
> constrain the treble to plain hunting and change the place notation, when it
> leads, to join courses together.
>
> Colin
>
>
> ----- Original Message ----- From: "Alexander Holroyd" <holroyd at math.ubc.ca>
> To: <ringing-theory at bellringers.net>
> Sent: Wednesday, April 27, 2011 1:19 AM
> Subject: Re: [r-t] ringing the Mathieu group
>
>
>> In a similar vein, does anyone know how to get an in-course extent of
>> caters using only the place notations 1, 5, and 9?
>>
>> On Tue, 26 Apr 2011, Alexander Holroyd wrote:
>>
>>> Here is today's brain teaser.
>>>
>>> Consider the three 12-bell place notations
>>> x 125T 18ET
>>>
>>> The group generated by these pns (i.e. the set of all rows you can get to
>>> from rounds using only these pns) contains 95040 rows. It is a very
>>> interesting group from a mathematical perspective, called the Mathieu
>>> Group M_12. (It is the second smallest of the 26 "sporadic groups"). One
>>> interesting property is that it is "sharply 5-transitive", which means
>>> that any given 5 bells (e.g. 12345) ring exactly once of each of the
>>> possible places that 5 bells can occupy (counting different orders of
>>> 12345 as different), giving 12x11x10x9x8 = 95040 rows.
>>>
>>> According to the "Lovasz conjecture", it should be possible to ring a true
>>> round block of these 95040 rows using only these three pns. Can anyone
>>> come up with an elegant way of doing this? It would obviously be nice to
>>> do it right-place, ie without 3 consecutive blows. I don't know whether
>>> that's possible.
>>>
>>> Ander
>>>
>>
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