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--- table_seating [2021-06-12 11:59] – created nik
+++ table_seating [2021-06-17 07:48] – timbo
@@ Line 2: / Line 2: @@
-arranging a group of people into a number of tables so that everyone sits with everyone else.
+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.
-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.
+  * 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
+Some general thoughts. Each sitting defines a partition of the set of people, each part is one table.
-Strict versions include 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)
+===Strict===
-  - https://en.wikipedia.org/wiki/Kirkman%27s_schoolgirl_problem
+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).
-  - https://oeis.org/search?q=schoolgirl&sort=&language=german&go=Suche
-In less strict cases we allow people to meet more often, or not to meet.
+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.
-The "Dagstuhl Happy Diner problem" is the version where everyone meets at least once.
-  - https://github.com/fpvandoorn/Dagstuhl-tables
-  - https://oeis.org/A318240
-Equitable Resolvable coverings also seem to be a more strict form, where we can allow people to meet at most twice.
+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]]}
-  - https://www.researchgate.net/publication/227715273_Equitable_resolvable_coverings
-  - https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.10024?saml_referrer
+=== 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]]}
+[[https://www.researchgate.net/publication/227715273_Equitable_resolvable_coverings|Equitable Resolvable coverings]] also seem to be a more strict form, where we can allow people to meet at most twice. {[[https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.10024|ref]]}
 If we have people sitting at round tables and only interacting with their neighbours, then we have the more difficult [[https://en.wikipedia.org/wiki/Oberwolfach_problem|Oberwolfach Problem]]