Timeline for order of a permutation and lexicographic order
Current License: CC BY-SA 4.0
18 events
| when toggle format | what | by | license | comment | |
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| Aug 25, 2018 at 19:25 | vote | accept | jcdornano | ||
| Aug 23, 2018 at 19:53 | answer | added | Claude Chaunier | timeline score: 1 | |
| Aug 23, 2018 at 13:35 | comment | added | Claude Chaunier | Now get ready for a shock. There is no such general truth for $Q=J$ and period $2$. For a few $6\times 7$ binary matrices, the period is $4$. For a few others, it is $6$. Some others, $3$. I'll show you those counterexamples in an answer. I'll also explore random big matrices to see better. | |
| Aug 23, 2018 at 13:31 | comment | added | Claude Chaunier | Great, I understand your explanation about $Id$. In other words : the first column never increases as a binary number, so it is going to stay the same at some point, as two blocks of $00\ldots0$ and $11\dots1$. At that point the same holds for the 2nd column for each sublock, and so on. | |
| Aug 22, 2018 at 20:08 | comment | added | jcdornano | I thing the case $Q=Id$ is true. We first notice that in the cycle there should appear a column (seen like the characteristic fonction of a set) that is a minimum for inclusion, and then it will state at the first place all the time. We do the same statement with the bloc of rows that begin by $0$ and those that begin by $1$, and so on... (dont have place to be very precise and formal^^) | |
| Aug 22, 2018 at 20:03 | comment | added | jcdornano | That's great, and very usefull! And I'm sure the list will teach us something! | |
| Aug 22, 2018 at 16:35 | comment | added | Claude Chaunier | I brute-force went through all binary matrices with $n=m\le6$, and it happens to holds for $Q=Id$ and $Q=J$ on those small sizes. It doesn't hold for many $Q$'s however. I'll list those for which it holds. | |
| Jul 23, 2018 at 19:19 | comment | added | jcdornano | I posted a new question with general $Q\in GL_m(F_2)$ such that $Q^q=Id$ here : mathoverflow.net/questions/306673/… | |
| Jul 23, 2018 at 15:41 | comment | added | jcdornano | It "seems" like it is also working for arbitrary $Q\in M_m(F_2)$ such that $Q^q=Iq$, and maybe we can have a nice generalization to any field and even to any ring ! (after deciding an arbitrary total order on the ring so that you can define a lexicographic order on rows/columns) | |
| Jul 22, 2018 at 21:27 | history | edited | jcdornano | CC BY-SA 4.0 |
added 75 characters in body
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| Jul 22, 2018 at 18:44 | comment | added | jcdornano | @Andreas Blass : Thank you very much and sorry for the wasted time! I edited , and I think I corrected all typos, I also fixed an example, to avoid misunderstanding from any typos or blunder I did't see | |
| Jul 22, 2018 at 18:37 | history | edited | jcdornano | CC BY-SA 4.0 |
annoying Typos!!
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| Jul 22, 2018 at 18:31 | history | edited | jcdornano | CC BY-SA 4.0 |
Typos
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| Jul 22, 2018 at 18:25 | history | edited | jcdornano | CC BY-SA 4.0 |
Thanks to Andreas Blass I realised that I had forget the composition with $\mathcal R$ !! I also add an example
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| Jul 22, 2018 at 17:45 | comment | added | jcdornano | I'm sorry I completely forgot a step in my construction (the composing with transposition part (that occurs in the link) is missing!!) I 'm editing write now!! | |
| Jul 22, 2018 at 14:12 | comment | added | Andreas Blass | I think I'm misunderstanding the question, because it seems to me that $r=1$ would work. Could you give me an example where $r$ can't just be $1$? I hope that such an example will clear up my confusion. | |
| Jul 22, 2018 at 4:44 | history | edited | jcdornano | CC BY-SA 4.0 |
typo
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| Jul 22, 2018 at 4:37 | history | asked | jcdornano | CC BY-SA 4.0 |