Timeline for Maximum conjugacy class size in $S_n$ with fixed number of cycles
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Jun 10, 2019 at 22:07 | comment | added | Aaron Meyerowitz | If one hypothesizes that the optimal solution for some range of $k$ is something like $1^a2^b3^c4\.5\.6\cdots \ell\.(n-x)$ then one could look at the largest classes subject to that restriction. For some conjectures and then try to prove them. | |
| Jun 9, 2019 at 10:11 | history | edited | Geoff Robinson | CC BY-SA 4.0 |
Suggested possible continuations when repeated parts were allowed.
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| Jun 7, 2019 at 19:31 | comment | added | Geoff Robinson | Yes, thanks Gerhard. I find it a little tricky to keep things under control when changes to a permutation introduce repeated cycle sizes which weren't there previously, so I can't really see the right generalization. | |
| Jun 7, 2019 at 18:08 | comment | added | Gerhard Paseman | This is similar to my local optimization suggestion. You might attempt a generalization as follows: group the cycle lengths so that there are d cycles of length less than l, and (let's assume c < d) c cycles of length l or greater, and that n and the cycles are large enough that we have room to play. Shorten each of the d cycles by 1 and assume we can distribute these d elements among the c cycles. I conjecture that the relevant product shrinks. If so, we can greedily optimize by each multiset of a_j. Gerhard "Is Omitting The Fine Print" Paseman, 2019.06.07. | |
| Jun 7, 2019 at 10:41 | history | edited | Geoff Robinson | CC BY-SA 4.0 |
tiny typo
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| Jun 7, 2019 at 10:35 | history | edited | Geoff Robinson | CC BY-SA 4.0 |
minor amendments
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| Jun 7, 2019 at 10:17 | history | answered | Geoff Robinson | CC BY-SA 4.0 |