[r-t] Bobs-only Stedman Triples - 12 complete B-block peals

[Andrew Johnson]

>Andrew Johnson writes,
> > These are the last of the bobs-only peals for now; I have some  
> > unpublished peals but they are not significantly different from the  
> > published ones.
> > The collection is here: https://complib.org/collection/11309
>This is amazing work, thank you Andrew. How comprehensive is this latest survey of yours? In other words, is
>it possible there are significant peals (in terms of interest of construction, number of bobs, ease of calling,
>etc) which are as yet undiscovered, or do you believe you have covered most possibilities?
>
>MBD
These are ad-hoc searches, and are not comprehensive. A lot of them were obtained by taking an existing peal I had found, keeping some parts fixed and varying other parts.

For example, sometimes I took one peal, bobbed all the Q-sets to get an odd number of round blocks, and kept the paired B-blocks as fixed because paired B-blocks are useful as there are a 6 ways each pair can be expressed, which helps linking everything together, and also if they were variable a search could come up with any of those 6 forms which wouldn't be significantly different, but would just flood the results.

I then took the remaining sixes and searched to see how many sets of round blocks I could obtain. I might also impose other limits like the total number of bobs (because otherwise the search might just turn up 84 B-blocks), and number of complete B-blocks, and also that any Q-sets should be bobbed (to restrict equivalent solutions). My searches couldn't easily filter results of odd numbers of round blocks vs even numbers so I had to do that once I had say 1000 results. If I got less than 1000 when the search terminated then I knew I had found all the blocks subject to the conditions. I could then try to link the blocks into a peal, but there would be a free choice of 6 for each of the paired B-blocks. Sometimes if I wanted to reduce the number of bobs I would choose the instances of the paired B-blocks to maximise the number of available Q-sets and then try to link the blocks without changing the types of the B-blocks. I could also try other restrictions such as not allowing 7 bobs in a row, and finding 8 plains in a row.

The number of bobs now ranges from 438 to 456 (existing 10-part peals) and 528 to 711 (one-part peals). With 84 B-blocks there are 840 bobs, so 123 omits on Q-sets could link those to 2 blocks with 717 blocks. An upper limit on the number of bobs therefore seems to be 714, but that would require a 2 blocks to 1 link with 3 omits, or a 4 blocks to 1 with 6 omits etc. which seems unlikely, so perhaps 711 bobs is the limit.

The 528 bob peal https://complib.org/composition/86761 had 4 Q-sets which could not be plained without splitting it back into multiple blocks, and there was another peal https://complib.org/composition/92761 where if all the Q-sets were plained there were 501 bobs (but multiple blocks), so perhaps a one-part peal using B-blocks can be found with less than 528 bobs but I'm guessing to achieve 500 bobs would be hard.

I found several peals with 8 plains in a row but they can't be arranged to put all at the start or all at the end of a peal with a normal start; it needs to start or end with a bob, so at best with my peals you can end with 8 plains and a bob, or 7 plains. Perhaps this can be improved on.

https://complib.org/composition/84174 is interesting as all the bobs come in multiples of 2 (2,4 or 6) except for 18 single bobs, but it is a one-part. The treble does 0,2 or 6 bobs behind. Compare with the exact 3-part from 2017 https://complib.org/composition/36006, which has 9 runs of 1 bob, 3 runs of 3 bobs and 15 runs of 5 bobs and the 7 does either 2 or 6 bobs behind.

Among the things that didn't work for me:
7-part peals
There is no bobs-only exact 7-part. There are multiple sets of blocks with a 7-part group with an odd number [5,21,35] of round blocks, but it doesn't seem possible to link them with bobs or omits. Here are some peals with singles to link the blocks

35 round blocks, 49 complete B-blocks, signature 63:14+105

3124567QS-----PPP--------PP----P---P---------P--------P---------PP---------P---*1(7)
4125367SQ-------P--P------P--*1(7)
4236175SQ----------*1(7)
5316472SQ----------*1(7)
1627354SQ----------*1(7)
https://complib.org/composition/48931 651 bobs, 42 singles

5 round blocks, 14 complete B-blocks, signature 84:5+252

3124567QS-----P-------PP-P-----PP--PP---------P--PP-----P-PPP----PP-PP-P-P-P-----P---*7(1)
6421375QS--PP----PP---P---P--------PP*7(1)
4125367SQ-----P*7(1)
1536472QSP-----*7(1)
7165324QSP-PP*7(1)
https://complib.org/composition/48932 560 bobs, 56 singles

35 round blocks, 49 complete B-blocks, signature 63:14+105

1234567QS--P--P--------P---------PP-------PP-P---------P--P---PPP--------P---------P-----*1(7)
4267351SQ----------*1(7)
1637452QS----------*1(7)
5316472SQ----------*1(7)
3617254SQ----------*1(7)
https://complib.org/composition/48954 651 bobs, 42 singles

6-part peals
There can't be an exact 6-part bobs-only peal, but the group can generate blocks which link in pairs to give 3 round blocks. When I was working with Dr Michael Haythorpe he ran my program in 2013 and we found the following sets of an odd number of blocks, shown in file odd.txt. Unfortunately I cannot see a way of linking the blocks with just bobs. That's an especial pity because the set giving 5 round blocks has only 402 bobs.

We also found some single blocks, shown in file single.txt, which when expanded for a peal formed 2 or 6 round blocks (but unfortunately not 3). This shows that the 6-part graph is Hamiltonian. Example peals using singles are here:

https://complib.org/composition/49158 540 bobs, 6 singles
https://complib.org/composition/49177 540 bobs, 6 singles

5-part peals
As already published, there just one instance (+rotations/reversal) of a single block (actually from the 10-part group, so made of two identical sections), but it links to itself and cannot be linked by Q-sets of bobs or omits. 
https://complib.org/composition/28949 450 bobs, 10 singles
There are sets of multiple blocks with odd numbers of blocks when expanded to all the rows (e.g. the original 10-part peals) which can be linked with Q-sets of bobs or omits to provide the existing 10-part peals.

The magic blocks appear in a 5-part search, together with the remaining B-blocks, but cannot be linked without changing the type of some of the B-blocks, so there isn't a 5-part based magic block peal. There might be other sets of an odd number of blocks - I haven't found them yet.

4-part peals - group [6.26]
Not much success here - but the search space gets larger as the groups get smaller.
The graph is Hamiltonian, but the example I found, from a 20-part search) divides into 4 1-part blocks. This doesn't stop there being an exact 4-part out there.
3124567QSP--PP--P---------P--PPP---------P--P--P--P*5(4)
Generators [1543267, 1425376] group order 4

4-part peals - group [4.04]
Not directly useful for a bobs-only peal.
This is the only group for Stedman Triples where I don't know whether the reduced graph is Hamiltonian i.e. can a search just turn up one block

4-part peals - group [6.35]
Not directly useful for a bobs-only peal. Is a Hamiltonian graph (i.e. search can find one block), but then need 4 of those blocks cover all the sixes.

4-part peals - group [4.05m]
An in course subgroup of the Thurstans' 20-part out of course group.
Not directly useful for a bobs-only peal. Is a Hamiltonian graph.

4-part peals - group [4.06m]
Two pairs swapping independently
Not directly useful for a bobs-only peal. Is Hamiltonian though, giving this nice peal with singles just swapping 12 or 34 in the parts:
https://complib.org/composition/66787 486 bobs, 6 singles

3-part peals
Lots of peals - but too big to search completely.
I think I did complete a search choosing 4 pairs of complementary B-blocks and one block with the remaining sixes (200 sixes), anyway it was how I found the 42 and 48 complete B-block exact 3-parts.

2-part peals - group [4.07]
Nothing new after the 2012 peals

2-part peals - group [2.01]
One pair swapping. Not directly useful for a bobs-only peal.
Is Hamiltonian though, and gives a peal which looks like Slack's two-part, but comes round half-way if you replace the single with a bob.
https://complib.org/composition/66793 508 bobs, 2 singles
Slack's peal is actually made of 4 blocks and does not come round if you replace the single with a bob or a plain and has more bobs (596 vs 508), so mine is cleaner, but not of the same historical interest.
https://complib.org/composition/29011 596 bobs, 2 singles

Andrew Johnson
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