Timeline for How many diagrams interlace a given Young diagram?
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 14, 2022 at 6:47 | comment | added | Per Alexandersson | Link to Gelfand-Tsetlin pattern definition: symmetricfunctions.com/gtpatterns.htm (and how they relate to SSYTs) | |
| Nov 8, 2022 at 17:49 | comment | added | David E Speyer | Oh, you are right. I can think of contexts in which I would be right, but the most straightforward thing is what @SamHopkins said. | |
| Nov 8, 2022 at 16:14 | comment | added | Sam Hopkins | @DavidESpeyer: Your observation about the product formula is of course correct. But regarding SSYT, I do think we want $\mu$ to have $n-1$ entries, as in the question-asker's post. For example, think of the way Gelfand-Tsetlin patterns are triangular: each row has one less entry than the previous one. | |
| Nov 8, 2022 at 16:08 | comment | added | David E Speyer | Isn't this $\prod_{j=1}^{n-1} (\lambda_j - \lambda_{j+1}+1)$? Because $\lambda_j - \lambda_{j+1}+1$ is the number of choices for $\mu_j$, and they can all be chosen independently. (PS If you want to exactly match SSYT, you should also allow $\mu$ to have an $n$-th part, with $\lambda_n \geq \mu_n \geq 0$.) | |
| Nov 8, 2022 at 15:53 | comment | added | Sam Hopkins | If we use $\mu \prec \lambda$ to denote "$\mu$ interlaces $\lambda$," then it is well-known that the number of semistandard Young tableaux of shape lambda with entries in $\{1,2,\ldots,n\}$ is the number of sequences $\lambda^{0} \prec \lambda^{1} \prec \cdots \prec \lambda^{n} = \lambda$. Each $\lambda^{i-1} \prec \lambda^{i}$ determines a horizontal strip of where the $i$'s in the SSYT are. You're basically asking for the "first step" of this process: the number of ways to place $n$ in an SSYT of shape $\lambda$. There should be a determinantal formula for this, at a minimum. | |
| Nov 8, 2022 at 15:38 | history | asked | Nicolas Medina Sanchez | CC BY-SA 4.0 |