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| table_seating [2024-04-11 10:13] – [Strict] nik | table_seating [2024-04-11 10:15] – [Strict] nik |
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| A strict version is an affine plane. Example 25 people in 5 tables of 5, Point set is $Z_5 \times Z_5$, 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 $Z_5 \times Z_5$, 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). |
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| 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. | 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. |
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| 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 $\iff v = 3 mod 6$. | 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 $\iff v = 3 mod 6$. |
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| For pairs, resolvable (v,2,1) 2-designs exist only for even v, v >= 4. | For pairs, resolvable $(v,2,1) 2\dash{}designs$ exist only for even v, v >= 4. |
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| Table size 4: Resolvable (v,k,1)- 2-designs. | Table size 4: Resolvable (v,k,1)- 2-designs. |
nik