Pairwise Balanced Designs with block size {4,5}

[Mahjong]

Hello,

I am searching small (4<v<24) Pairwise Balanced Designs with block sizes 4 and 5.

I already found that v must be 0,1 mod (4), but not equal to 8, 9 or 12. So,for instance, PBD-(13,{4,5}) should exist.

There is a paper titled "Pairwise Balanced Designs with Consecutive Block Sizes" by ACH Ling, but it seems to only deal with block size 8,9 and 10.

Any pointers would be great.

[Ian Wakeling]

I don't think I can help very much. I have not read the paper that you site and have not studied PBDs in any detail.

Why do you need to have blocks of size 5 for PBD-(13,{4,5}), when there is a BIBD-(13,4,lambda=1) that you can use instead?

I note that BIBDs (21,5,1) and (25,5,1) also exist.  So deleting any one symbol from these would lead to designs with k={4,5}.

Adding a symbol to resolvable BIBD-(16,4,1) is another possibility.

[Mahjong]

Why do you need to have blocks of size 5 for PBD-(13,{4,5}), when there is a BIBD-(13,4,lambda=1) that you can use instead?

I am trying to list all possible designs useful for small Mahjong clubs, like the one I have founded.

For tournaments, BIBD-(13,4,1) could almost work. BIBD-(13,4,3) is a lot better because we can have everybody (minus one player who can act as the tournament director for that session) compete on three simultaneous tables. That's for tournaments.

For regular meetings, I am also searching for designs. We are often short a few players from 0 mod 4. When it happens, we allow tables with five players. So we often have a mix of {4,5} tables.  I want, as much as possible, that all members of the club meet each other. I am trying to find PBD designs to see if they could help.

I note that BIBDs (21,5,1) and (25,5,1) also exist.  So deleting any one symbol from these would lead to designs with k={4,5}.

I think you might have given me a very valuable hint! Thank you very much.

I will try to convert a BIBD(21,5,1) into a PBD and see how it works out.

[Ian Wakeling]

Be careful as PBDs are not necessarily what you want.  There is no requirement for them to be resolvable.  Also there is no requirement that they are equireplicate, so different players may have different numbers of games.

I had some off-line discussions with another contact regarding mahjong schedules, they were willing to run tables with only 3 players:

 1 20  9 22 | 4  3 13  6 | 14  8 10 15 | 19  2 16 17 | 23  5 21 11 | 12  7 18 
 2 21 10 22 | 5  4 14  7 |  8  9 11 16 | 20  3 17 18 | 23  6 15 12 | 13  1 19 
 3 15 11 22 | 6  5  8  1 |  9 10 12 17 | 21  4 18 19 | 23  7 16 13 | 14  2 20 
 4 16 12 22 | 7  6  9  2 | 10 11 13 18 | 15  5 19 20 | 23  1 17 14 |  8  3 21 
 5 17 13 22 | 1  7 10  3 | 11 12 14 19 | 16  6 20 21 | 23  2 18  8 |  9  4 15 
 6 18 14 22 | 2  1 11  4 | 12 13  8 20 | 17  7 21 15 | 23  3 19  9 | 10  5 16 
 7 19  8 22 | 3  2 12  5 | 13 14  9 21 | 18  1 15 16 | 23  4 20 10 | 11  6 17

Is that sort of thing any good?

[Mahjong]

This is perfect. Resolvable and equireplicate. And, yes, {3,4} seem more appropriate : a majority of foursomes and only a few threesomes. While with {4,5}, it seems that there are almost only fivesomes and only a few foursomes.
I