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table_seating [2021-12-08 08:44] – timbo | table_seating [2024-04-11 10:13] – [Strict] nik | ||
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==== Table Seatings ==== | ==== Table Seatings ==== | ||
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The general problem is arranging a group of people into a number of tables so that everyone sits with everyone else. There are multiple versions for this. | The general problem is arranging a group of people into a number of tables so that everyone sits with everyone else. There are multiple versions for this. | ||
- | * The strict version is that all tables are the same size and that after the required number of rounds, everyone has shared a table with every other person exactly once | + | * The **strict** version is that all tables are the same size and that after the required number of rounds, everyone has shared a table with every other person exactly once |
- | * The lower version requires that each person shares a table with each other person at most once | + | * The **lower version** requires that each person shares a table with each other person at most once |
- | * The upper version requires that each person shares a table with each other person at least once | + | * The **upper version** requires that each person shares a table with each other person at least once |
Some general thoughts. Each sitting defines a partition of the set of people, each part is one table. | Some general thoughts. Each sitting defines a partition of the set of people, each part is one table. | ||
- | ===Strict=== | + | ====Strict==== |
- | A strict version is an affine plane. Example 25 people in 5 tables of 5, Point set is Z_5 x Z_5,we take the tables to be the lines L(a, | + | A strict version is an affine plane. Example 25 people in 5 tables of 5, Point set is $Z_5 \times |
More generally we want a [[https:// | More generally we want a [[https:// | ||
- | Versions include [[https:// | + | Versions include [[https:// |
For pairs, resolvable (v,2,1) 2-designs exist only for even v, v >= 4. | For pairs, resolvable (v,2,1) 2-designs exist only for even v, v >= 4. | ||
Table size 4: Resolvable (v,k,1)- 2-designs. | Table size 4: Resolvable (v,k,1)- 2-designs. | ||
- | [[https:// | + | [[https:// |
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- | + | ||
- | === Lower Version=== | + | ==== Lower Version==== |
Just leave out some sittings on a strict version. Perhaps add a few nonexistant people to get a better distribution of people on tables with not all tables always full. | Just leave out some sittings on a strict version. Perhaps add a few nonexistant people to get a better distribution of people on tables with not all tables always full. | ||
- | === Upper Version === | + | ==== Upper Version |
The " | The " |