Timeline for What are the periodic Dyck paths?
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| Dec 20, 2019 at 10:20 | history | edited | Mare | CC BY-SA 4.0 |
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| Oct 15, 2018 at 20:21 | vote | accept | Mare | ||
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| Sep 28, 2018 at 17:17 | history | edited | Mare | CC BY-SA 4.0 |
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| Sep 28, 2018 at 17:00 | answer | added | Mare | timeline score: 1 | |
| S Sep 28, 2018 at 16:40 | history | bounty ended | Mare | ||
| S Sep 28, 2018 at 16:40 | history | notice removed | Mare | ||
| Sep 26, 2018 at 21:55 | answer | added | Gjergji Zaimi | timeline score: 7 | |
| Sep 21, 2018 at 11:55 | comment | added | Mare | @Vincent This is explained for example here findstat.org/DyckPaths/NakayamaAlgebras (to be updated soon). | |
| Sep 21, 2018 at 11:33 | comment | added | Vincent | Really sorry to bother you again, but can you explain the link between the 'standard' notation for Dyck paths and your notation? (At some point I had convinced myself it was obvious, but later it turned out I was totally mistaken and now I can't seem to find the correct link between the two.) | |
| Sep 21, 2018 at 11:07 | comment | added | Mare | @Vincent that should be wrong, here you can find some values: findstat.org/StatisticsDatabase/St000684 | |
| Sep 21, 2018 at 11:04 | comment | added | Vincent | Can I check something first? I know nothing about the algebra-side of things, but is this global dimension you talk about by any chance equal to the maximum allowed 'jump' $c_{i+1} - c_i$? | |
| Sep 21, 2018 at 10:54 | comment | added | Vincent | Ok, I will try and see if I can make it a bit more concrete first. | |
| Sep 21, 2018 at 10:52 | comment | added | Mare | @Vincent Maybe answer it as an answer where upvotes are not possible (community wiki?). I have an explanation now for the fibo, tribo,.... numbers as I understand the global dimension of the algebras corresponding to bouncing Dyck paths. Proving the conjecture in the current form of the thread would also prove the previous conjectures about the fibo, tribo.... numbers. | |
| Sep 21, 2018 at 10:49 | comment | added | Vincent | Reading through some older versions of your post (thanks for going through the trouble of all the edits BTW) where you mention fibo- and tribo and pentonnacci numbers I have a vague feeling this relates to something I have been studying before, but I don't understand the connection at all and it is a bit mushy in my head. Would you mind if I typed it up and send it to you so we can perhaps see if we can make sense of it together? I expect it to be too long for a comment and too non-concrete to be an answer but perhaps MO has some sort of personal messaging service? | |
| Sep 21, 2018 at 10:39 | comment | added | Mare | @Vincent No, it seems to be very complicated. But "bouncing Dyck path" has a nice visual interpretation as you might see when looking at some examples. (you can click the findstat link to see pictures of the Dyck paths and look for the period to filter the bouncing ones out). | |
| Sep 21, 2018 at 10:31 | comment | added | Vincent | Do you know if the notion of periodicity has a nice 'visual' interpretations in terms of the staircase walks the wolfram-site is talking about? | |
| Sep 21, 2018 at 10:28 | comment | added | Vincent | O hey, I see now that you edited in the link into your comment. Thanks! This was just what I was looking for | |
| Sep 21, 2018 at 10:26 | comment | added | Vincent | By "ALL Dyck path" I meant "ALL Dyck paths". I forgot to type the "s". And I see now that the answer to my question was already in your post and I had glossed over it. | |
| Sep 21, 2018 at 10:24 | history | edited | Mare | CC BY-SA 4.0 |
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| Sep 21, 2018 at 10:22 | comment | added | Mare | @Vincent What do you mean with "ALL Dyck path"? They are enumerated by the Catalan numbers as is well known and as I wrote in the beginning of the thread ( see mathworld.wolfram.com/DyckPath.html ), but the periodic ones are the interesting ones and not all Dyck paths are periodic as can be seen in the two examples. | |
| Sep 21, 2018 at 10:20 | comment | added | Vincent | Maybe a silly question but since you ask about how to enumerate the periodic paths, is it trivial/easy/hard/impossible to enumerate ALL Dyck path? And what is the outcome? | |
| Sep 21, 2018 at 10:19 | history | edited | Mare | CC BY-SA 4.0 |
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| Sep 21, 2018 at 10:08 | history | edited | Mare | CC BY-SA 4.0 |
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| Sep 21, 2018 at 9:59 | history | edited | Mare | CC BY-SA 4.0 |
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| S Sep 21, 2018 at 9:54 | history | bounty started | Mare | ||
| S Sep 21, 2018 at 9:54 | history | notice added | Mare | Draw attention | |
| Sep 21, 2018 at 9:53 | history | edited | Mare | CC BY-SA 4.0 |
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| Jun 28, 2018 at 17:27 | comment | added | Robert Frost | This is just a suggestion but a relation may appear more clearly if you filter those periodic sequences into "prime" sequences only; i.e. include only Dyck sequences which are not a repetition of some shorter sequence. | |
| Jun 23, 2018 at 23:20 | history | edited | Mare | CC BY-SA 4.0 |
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| Jun 23, 2018 at 21:02 | comment | added | Mare | @MichaelAlbert Thanks, that sounds interesting. I will look into this. Probably an easy description of those Dyck paths will make it easy to prove that they have those properties, but the other direction might be harder (namely that there are no other). | |
| Jun 23, 2018 at 20:39 | comment | added | Michael Albert | I know nothing about the topic but on the basis of the numbers and conjectures alone it would seem that one might be looking for a correspondence with compositions of n-1 (of which there are 2^{n-2}) with the various restrictions corresponding to restricting the size of the largest part (Fibonacci = largest part of size at most 2 etc.) The data seem to support the first interpretation with the Dyck paths of period (n+1) apparently being exactly those that are simply the concatenation of a series of peaks i.e., concatenations of parts 1^k 0^k for some k > 0. | |
| Jun 23, 2018 at 19:34 | history | edited | Mare | CC BY-SA 4.0 |
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| Jun 23, 2018 at 19:21 | history | edited | Mare | CC BY-SA 4.0 |
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| Jun 23, 2018 at 19:14 | history | edited | Mare | CC BY-SA 4.0 |
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| Jun 23, 2018 at 18:40 | history | edited | Mare | CC BY-SA 4.0 |
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| Jun 23, 2018 at 18:31 | history | asked | Mare | CC BY-SA 4.0 |