Timeline for equivalence classes of arch diagrams in bijection with permutations
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 8, 2016 at 20:30 | vote | accept | Noam Zeilberger | ||
| Sep 8, 2016 at 19:24 | comment | added | benblumsmith | @NoamZeilberger - I thought about it more and don't think it quite works. The translation I had in mind is not bijective from normalized arch diagrams to permutations. Ilya Bogdanov's answer seems to answer question 2 though. | |
| Sep 8, 2016 at 19:10 | answer | added | BWW | timeline score: -1 | |
| Sep 8, 2016 at 17:50 | answer | added | Ilya Bogdanov | timeline score: 7 | |
| Sep 8, 2016 at 14:26 | comment | added | Noam Zeilberger | @SamHopkins thanks for the reference. I don't know enough about hyperplane arrangements to see the connection, but I'll let it stew :-) | |
| Sep 8, 2016 at 14:24 | comment | added | Noam Zeilberger | @benblumsmith I don't quite understand your suggestion, though I share the intuition that it might be possible to read the normalized arch diagrams as encoding cycle decompositions of permutations. It also appears that the two-variable generating function $A(x,z)$ is the bivariate OGF for unsigned Stirling numbers of the first kind, which seems relevant. | |
| Sep 8, 2016 at 4:18 | comment | added | Sam Hopkins | I feel like this might be related to hyperplane arrangements, and in particular the following paper: arxiv.org/abs/1604.06554 | |
| Sep 8, 2016 at 3:05 | comment | added | benblumsmith | I unfortunately don't have time to check out all the details but it appears to me that the normalized arch diagrams in your picture above can be interpreted as the cycle diagrams of permutations on $n$ points, by thinking of each adjacent pair of dots, starting from the left, as a single node, and following the arches around the nodes starting with the first one (i.e. the one incident to the first point). This would answer question 2 if the details can be worked out. | |
| Sep 8, 2016 at 2:02 | history | edited | Noam Zeilberger | CC BY-SA 3.0 |
use $n$ for index
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| Sep 8, 2016 at 1:55 | history | asked | Noam Zeilberger | CC BY-SA 3.0 |