# Differences

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table_seating [2021-06-12 12:02] – nik | table_seating [2022-04-04 03:32] (current) – nik | ||
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==== Table Seatings ==== | ==== Table Seatings ==== | ||

+ | 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. | ||

- | arranging a group of people into a number of tables so that everyone sits with everyone else. | + | * 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 **upper version** requires that each person shares a table with each other person at least once | ||

- | A strict version is an affine plane. More generally we want a [[https://the parallelism. Maybe there is something like discrete hyperbolic geometry to deal with this, but we seem to have better combinatorial ideas below. | + | Some general thoughts. Each sitting defines a partition of the set of people, each part is one table. |

+ | ====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, | ||

- | Strict versions include [[https://Kirkman%27s_schoolgirl_problem|Kirkman']] (15 children walk in groups of 3, can they do this so that all pairs of girls walk together exactly once over a whole week) {[[https://.org/ | + | More generally we want a [[https://Block_design#|resolvable 2-design]]. We want resolvable (v,k,1) designs. Resolvable is the parallelism. Maybe there is something like discrete hyperbolic geometry to deal with this, but we seem to have better combinatorial ideas below. |

- | In less strict cases we allow people to meet more often, or not to meet. | + | |

+ | Versions include [[https://to resolvable (v,3,1) 2-designs, which if I understand it, exist iff v = 3 mod 6. | ||

+ | | ||

+ | For pairs, resolvable (v,2,1) 2-designs exist only for even v, v >= 4. | ||

+ | | ||

+ | Table size 4: Resolvable (v,k,1)- 2-designs. | ||

+ | [[https://not concern us. | ||

+ | | ||

+ | ==== 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. | ||

+ | | ||

+ | ==== Upper Version ==== | ||

The " | The " | ||

[[https:// | [[https:// | ||

+ | |||

If we have people sitting at round tables and only interacting with their neighbours, then we have the more difficult [[https:// | If we have people sitting at round tables and only interacting with their neighbours, then we have the more difficult [[https:// | ||