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 table_seating [2021-12-08 08:44] – timbo table_seating [2022-04-04 03:32] (current) – nik 2022-04-04 03:32 nik 2021-12-08 08:44 timbo 2021-06-17 07:48 timbo 2021-06-12 12:03 nik 2021-06-12 12:02 nik 2021-06-12 12:02 nik 2021-06-12 11:59 nik created 2022-04-04 03:32 nik 2021-12-08 08:44 timbo 2021-06-17 07:48 timbo 2021-06-12 12:03 nik 2021-06-12 12:02 nik 2021-06-12 12:02 nik 2021-06-12 11:59 nik created Line 1: Line 1: ==== 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. 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,b)={(x,y)| y=ax+b} and L(a)={(a,y)| y in Z_5}, the sitting is a parallel class (the 5 lines with the same slope a), so we have 6 sittings, L(a,b) for a=0,1,2,3,4 and then the parallel class of L(a). 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,b)={(x,y)| y=ax+b} and L(a)={(a,y)| y in Z_5}, the sitting is a parallel class (the 5 lines with the same slope a), so we have 6 sittings, L(a,b) for a=0,1,2,3,4 and then the parallel class of L(a). Line 23: Line 22: [[https://www.semanticscholar.org/paper/The-spectrum-of-resolvable-designs-with-block-size-Vasiga-Furino/364fb4a75a38493ed2c86fa3589adfee6d2714f5|This paper]] says that nessesary numerical conditions are sufficient except for a case that need not concern us. [[https://www.semanticscholar.org/paper/The-spectrum-of-resolvable-designs-with-block-size-Vasiga-Furino/364fb4a75a38493ed2c86fa3589adfee6d2714f5|This paper]] says that nessesary numerical conditions are sufficient except for a case that need not concern us. - + ==== 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 "[[https://github.com/fpvandoorn/Dagstuhl-tables|Dagstuhl Happy Diner problem]]" is the version where everyone meets at least once. {[[https://oeis.org/A318240|oeis]]} The "[[https://github.com/fpvandoorn/Dagstuhl-tables|Dagstuhl Happy Diner problem]]" is the version where everyone meets at least once. {[[https://oeis.org/A318240|oeis]]}
• table_seating.txt