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 table_seating [2021-06-12 12:03] – nik 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. - 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://en.wikipedia.org/wiki/Block_design#Resolvable_2-designs|resolvable 2-design]]. 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. + 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,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). - Strict versions include [[https://en.wikipedia.org/wiki/Kirkman%27s_schoolgirl_problem|Kirkman's Schoolgirl Problem]] (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://oeis.org/search?q=schoolgirl&sort=&language=german&go=Suche|oeis]]} + More generally we want a [[https://en.wikipedia.org/wiki/Block_design#Resolvable_2-designs|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://en.wikipedia.org/wiki/Kirkman%27s_schoolgirl_problem|Kirkman's Schoolgirl Problem]] (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://oeis.org/search?q=schoolgirl&sort=&language=german&go=Suche|oeis]]}. This is the question as 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://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==== + + 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 "[[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
• Last modified: 2022-04-04 03:32
• by nik