Timeline for What is this numerically-generated function?
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12 events
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| Feb 17, 2022 at 5:57 | comment | added | eigengrau | @SamHopkins Sure!!!... Dyck paths, now that I look at your refernce (also math.mit.edu/~apost/courses/18.204-2016/… and a million other google hits). And also Max's Lattice words. Wonder if the poster of the original math.stackexchange.com/questions/4380919 question knew any of that. I certainly didn't, but it somehow seemed interesting enough to pursue. And it seems lots of people have already done that under various different rubrics. Even, as Max points out, for "more than 2 letters". Thanks, everybody, for the elaborations. | |
| Feb 16, 2022 at 19:25 | comment | added | Max Alekseyev | @SamHopkins: en.wikipedia.org/wiki/Lattice_word ? | |
| Feb 16, 2022 at 19:14 | comment | added | Sam Hopkins | A perhaps more interesting/fruitful way to extend the problem to more than $2$ letters would be: words in $1,2,...,n$ such that for every prefix, $\#$ of $1$'s is greater than $\#$ of $2$'s is greater than $\#$ of $3$'s etc. With the further restriction that the number of each letter is the same, then you're looking at Standard Young Tableaux of rectangular shape (i.e., "multi-dimensional Catalan numbers"). | |
| Feb 16, 2022 at 16:38 | comment | added | Peter Taylor | The stuff in my profile relates to a series of bad decisions by Stack Exchange (the company) made far worse by their handling of the community response. The "Monica incident" was the most notable. I've been boycotting the SE network since about Christmas 2019, but I except MO from the boycott because it's run by a separate foundation. | |
| Feb 16, 2022 at 15:59 | comment | added | Sam Hopkins | The "base 2" case are called "Dyck path prefixes": see e.g. arxiv.org/abs/1406.1709. | |
| Feb 16, 2022 at 15:38 | history | edited | Max Alekseyev |
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| Feb 16, 2022 at 15:34 | vote | accept | eigengrau | ||
| Feb 16, 2022 at 15:32 | answer | added | Max Alekseyev | timeline score: 1 | |
| Feb 16, 2022 at 14:51 | comment | added | eigengrau | @PeterTaylor Thanks, I guess it's maybe not as interesting a relationship as I'd originally thought. (P.S. what's that "losing confidence" stuff in your profile about?) | |
| Feb 16, 2022 at 14:36 | comment | added | Peter Taylor | There's an obvious relationship between the titles of the OEIS sequences you link for bases 3 and 4, and if you look further down on the page for base 5 you'll see the corresponding entry in the "formula" section: for base $b$ the generating function appears to be $$\frac{c((b-1)x^2)}{1 - xc((b-1)x^2)} = \frac{1 - \sqrt{1-4(b-1)x^2}}{2(b-1)x^2 - x + x\sqrt{1-4(b-1)x^2}}$$ | |
| S Feb 16, 2022 at 14:23 | review | First questions | |||
| Feb 16, 2022 at 14:36 | |||||
| S Feb 16, 2022 at 14:23 | history | asked | eigengrau | CC BY-SA 4.0 |