Timeline for Distribution of peaks in Dyck paths
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Nov 8, 2023 at 6:45 | comment | added | Per Alexandersson | Put q=0 in Thm. 3.4 in arxiv.org/pdf/1405.6919.pdf, and you get Narayana, and a connection with ASEP. This might be something easier to study | |
| Nov 7, 2023 at 16:11 | comment | added | David E Speyer | Cohn, Larsen and Propp computed the limiting shape of a plane partition in the box of size $(\alpha n) \times (\beta n) \times (\gamma n)$; it is is frozen outside an inscribed ellipse. nyjm.albany.edu/j/1998/4-10.pdf . You want plane partitions inside $2 \times (k-1) \times (n-k)$, which we can think of as $(\alpha, \beta, \gamma) = (0, r, 1-r)$. I'm not sure that their paper directly applies when one of the rescaled side lengths limits to zero, but it tells me which way to bet. | |
| Nov 7, 2023 at 16:08 | comment | added | David E Speyer | This time we have two paths which are constrained to be noncrossing. But, if the rescaled paths ever get a constant distance apart, the previous analysis will take over and make them both become line segments. So I think the optimum will just be that both paths limit to the diagonal of the rectangle, and the peaks are equidistributed. | |
| Nov 7, 2023 at 16:06 | comment | added | David E Speyer | If you look at a single partition in the $k \times (n-k)$ box, as $n \to \infty$ with $k/n \to r$, it is almost certain that the path will limit to the straight line. This is because the entropy function $\lim_{n \to \infty} \tfrac{1}{n} \log \binom{n}{pn} = - p \log p - (1-p) \log (1-p)$ is convex in $p$, so it is optimal to have the curve take the same slope everywhere. I wrote this up in a blogpost a long time ago sbseminar.wordpress.com/2011/10/02/random-partitions-i . (continued) | |
| Nov 7, 2023 at 16:02 | comment | added | David E Speyer | I suspect that equidistribution is almost certain. Here is a sketch of a proof: In @SamHopkins's notation, put $\lambda_i = \alpha_1+\cdots+ \alpha_i$ and $\mu_i = \beta_1 + \cdots + \beta_i$. Then Narayana numbers correspond to pairs of partitions $\lambda \subset \mu \subset [k-1] \times [n-k]$. So, sending $n \to \infty$ with $k/n \to r$, one can rescale $\lambda$ and $\mu$ and imagine them as noncrossing paths from $(0,0)$ to $(r,1-r)$, and one wants to know what the most likely such path is. (continued) | |
| Nov 7, 2023 at 15:27 | history | edited | Ben Deitmar | CC BY-SA 4.0 |
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| Nov 7, 2023 at 15:19 | history | edited | Ben Deitmar | CC BY-SA 4.0 |
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| Nov 7, 2023 at 15:03 | history | edited | Ben Deitmar | CC BY-SA 4.0 |
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| Nov 7, 2023 at 14:52 | comment | added | Sam Hopkins | Here's another comment: there is a straightforward bijection between Dyck paths of length $2n$ with $k$ peaks, and pairs $\alpha=(\alpha_1,\ldots,\alpha_k)$, $\beta=(\beta_1,\ldots,\beta_k)$ of compositions of $n$ into $k$ parts for which $\alpha_1 + \alpha_2 + \cdots + \alpha_i \geq \beta_1 + \beta_2 + \cdots + \beta_i$ for all $i$. We simply record the lengths of the up (respectively, down) run in $\alpha$ (resp., $\beta$). For example, in your pictured Dyck path we have $\alpha = (2,2,1)$ and $\beta=(1,3,1)$. This could be a useful alternative perspective for your problem. | |
| Nov 7, 2023 at 14:15 | review | Close votes | |||
| Nov 7, 2023 at 18:23 | |||||
| Nov 7, 2023 at 12:53 | history | edited | Ben Deitmar | CC BY-SA 4.0 |
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| Nov 7, 2023 at 12:43 | history | asked | Ben Deitmar | CC BY-SA 4.0 |