Timeline for Are There Always Group Generators Which Give Unimodal Growth?
Current License: CC BY-SA 3.0
29 events
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| Feb 14, 2016 at 3:34 | comment | added | David S. Newman | @AnthonyLabarre Yes. I believe that these permutations of length 9 were the shortest permutations that I found which give non-unimodal growth. However, I don't think that I did an exhaustive search. I think I took arbitrary permutations and then calculated the counting function for these pairs. | |
| Feb 8, 2016 at 10:52 | comment | added | Anthony Labarre | @DavidS.Newman Yes, this allowed me to find the flaw in my script, and the sequence I obtain is now identical to yours, thank you! Is $n=9$ the smallest value for which you found a counterexample? | |
| Feb 7, 2016 at 15:44 | comment | added | David S. Newman | @AnthonyLabarre I couldn't wait and did the calculations by hand. For words of length 4, (which is the first case where we disagree) using "a" for one permutation and "b" for the other, I find that aaaa = bbbb = the identity, while aabb = bbaa. Then, by my count, there are 16-3 = 13 words of length 4. Is this of some help? | |
| Feb 7, 2016 at 1:31 | comment | added | David S. Newman | @Anthony Labarre I am away from my computer for the next few days. Hopefully when I get back I'll be able to find my computations in my Mathematica files. If I have to redo the computations it will take me a substantially longer time. | |
| Feb 5, 2016 at 15:34 | comment | added | Anthony Labarre | The counting function I obtain for the generators you give is $(1, 2, 4, 8, 14, 23, 37, 52, 69, 98, 120, 132, 114)$, which is unimodal. Can you get in touch so we can figure out who's right and why? | |
| S Feb 10, 2014 at 22:19 | history | bounty ended | David S. Newman | ||
| S Feb 10, 2014 at 22:19 | history | notice removed | David S. Newman | ||
| Feb 5, 2014 at 17:44 | answer | added | Andy Wilson | timeline score: 3 | |
| S Feb 4, 2014 at 3:50 | history | bounty started | David S. Newman | ||
| S Feb 4, 2014 at 3:50 | history | notice added | David S. Newman | Draw attention | |
| Feb 4, 2014 at 2:12 | history | edited | Ricardo Andrade | CC BY-SA 3.0 |
fixed latex problems created by previous edit (I don't think this question necessarily had to have tex, but please try to maintain some tex-like coherence when you add latex); minor formatting; added question mark to title
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| S Feb 4, 2014 at 1:20 | history | suggested | user5794 | CC BY-SA 3.0 |
added LaTex
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| Feb 4, 2014 at 1:10 | review | Suggested edits | |||
| S Feb 4, 2014 at 1:20 | |||||
| Feb 4, 2014 at 1:07 | history | edited | Lee Mosher |
edited tags
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| Feb 4, 2014 at 0:48 | comment | added | David S. Newman | @YvesCornulier I've changed the definition of unimodal to correspond to that in MathWorld. Thanks. | |
| Feb 4, 2014 at 0:42 | history | edited | David S. Newman | CC BY-SA 3.0 |
several people commented that my definition of unimodal was not clear so I changed it citing the definition in MathWorld.
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| Dec 15, 2013 at 10:08 | comment | added | YCor | @David: no, it just means you forgot the word "local" in your definition. Then your definition coincides with Wikipedia's (which concerns sequences, otherwise I wouldn't have mentioned it). | |
| Dec 15, 2013 at 9:46 | comment | added | Christian Stump | Your definition above does not talk about local maxima, but about a global maximum. @YvesCornulier sequence does thus fit your definition of unimodality. | |
| Dec 15, 2013 at 3:19 | comment | added | David S. Newman | @YvesCornulier By my definition 1,2,1,3,3,0,0,0,0,.... is not unimodal. It has two local maxima:2 and 3. The sequence 1,2,3,3,0,0,0,... is unimodal by my definition. It has a single maximum: 3, although 3 appears more than once, its appearances come for successive values of f. The sequence 1,2,3,2,3,... would not meet my definition of unimodal. Although the only maximum is 3, its two appearances are separated by the value 2. I haven't looked at the Wikipedia definition, but I'd guess that it deals with functions on a continuous domain. Here we have values for f only at integers. | |
| Dec 15, 2013 at 2:05 | comment | added | YCor | @David: your definition of unimodal is not the same as in Wikipedia. Is the sequence $(1,2,1,3,3,0,0,0,0,\dots)$ unimodal for your definition (which confuses me: do you mean that the set on which $f$ is maximal is an integral interval? no monotonous behavior as in Wikipedia's definition?) | |
| Dec 15, 2013 at 0:03 | comment | added | David S. Newman | @Benjamin Steinberg I've added a comment that the conjecture is trivially true if k is taken to be the number of elements in the group. | |
| Dec 15, 2013 at 0:00 | history | edited | David S. Newman | CC BY-SA 3.0 |
I added a comment in agreement with one of the comments that the conjecture is trivially true for some values of k.
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| Dec 13, 2013 at 5:31 | comment | added | David S. Newman | @Douglas Zare I've added an example of a group and two of its generators for which the counting function is not unimodal. | |
| Dec 13, 2013 at 5:23 | history | edited | David S. Newman | CC BY-SA 3.0 |
I added an example of non-unimodal growth in response to a comment made by one of the viewers.
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| Dec 12, 2013 at 13:56 | comment | added | Benjamin Steinberg | Should k always be the minimal number of generators? If you use the whole group as generators than trivially you have unimodality. | |
| Dec 12, 2013 at 11:42 | comment | added | Christian Stump | For a finite Coxeter group of rank $n$ (crystallographic or not) with simple generators $S$ and reflections $T$, the length generating functions are known to be $(1+q+\ldots+q^{e_1}) \cdots (1+q+\ldots+q^{e_n})$ and $(1+e_1q)\cdots(1+e_nq)$ where $e_i$ are the exponents of the group. This implies unimodality in both cases. | |
| Dec 12, 2013 at 9:39 | comment | added | Qiaochu Yuan | In the special case of finite Coxeter groups with their standard generators unimodality should be well-known, e.g. for the Weyl groups it follows from hard Lefschetz applied to flag varieties. | |
| Dec 12, 2013 at 8:01 | comment | added | Douglas Zare | What is an example of a group and set of generators with nonunimodal growth? | |
| Dec 12, 2013 at 4:54 | history | asked | David S. Newman | CC BY-SA 3.0 |