Timeline for Identity involving Schur polynomials, binomial coefficients and contents of partition
Current License: CC BY-SA 4.0
19 events
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| Feb 4, 2022 at 0:11 | history | edited | Marcel | CC BY-SA 4.0 |
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| S Jan 29, 2022 at 17:06 | history | bounty ended | CommunityBot | ||
| S Jan 29, 2022 at 17:06 | history | notice removed | CommunityBot | ||
| S Jan 21, 2022 at 15:20 | history | bounty started | Marcel | ||
| S Jan 21, 2022 at 15:20 | history | notice added | Marcel | Draw attention | |
| Jan 21, 2022 at 13:15 | history | edited | Marcel | CC BY-SA 4.0 |
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| Jan 20, 2022 at 23:34 | history | edited | Marcel | CC BY-SA 4.0 |
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| Jan 20, 2022 at 23:15 | history | edited | Marcel | CC BY-SA 4.0 |
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| Jan 20, 2022 at 15:27 | history | edited | Marcel | CC BY-SA 4.0 |
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| Jan 19, 2022 at 15:47 | comment | added | Richard Stanley | @Marcel: there are many known identities involving $f^{\mu/\lambda}$. Perhaps one of them is relevant to your question. | |
| Jan 19, 2022 at 13:20 | history | edited | Marcel | CC BY-SA 4.0 |
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| Jan 18, 2022 at 22:28 | comment | added | Marcel | @FedorPetrov I am not aware of any such representation, no | |
| Jan 18, 2022 at 22:27 | comment | added | Marcel | @SamHopkins Yes, I am aware of that formula, it's what I used to verify the conjecture, using in Maple, for several $\lambda$ | |
| Jan 18, 2022 at 22:24 | comment | added | Marcel | @RichardStanley I see that $C_{\lambda\mu}$ can be written in terms of $f^{\mu/\lambda}$, which is the number of standard Young tableaux of shape $\mu/\lambda$. This seems more like an interpretation than a simplification... unless there is an independent way of computing $f^{\mu/\lambda}$ | |
| Jan 18, 2022 at 21:31 | comment | added | Richard Stanley | The determinant $C_{\lambda\mu}$ can be simplified. See problem 87 of klein.mit.edu/~rstan/ec/ch7suppsol.pdf. Put $n=0$ in the definition of $d_{\lambda\mu}$. | |
| Jan 18, 2022 at 20:46 | comment | added | user61318 | Yeliussizov discusses (a vast generalization) of your function in here: arxiv.org/pdf/1601.01581.pdf. In later work, he has considered probabilistic things (arxiv.org/abs/1907.06985 and arxiv.org/pdf/2002.10086.pdf). I would look into these. | |
| Jan 18, 2022 at 20:41 | comment | added | Sam Hopkins | Surely you are aware of this, but there is a formula for the evaluation $s_{\mu}(1^N)$: namely, Stanley's hook-content formula. | |
| Jan 18, 2022 at 20:19 | comment | added | Fedor Petrov | A natural approach would be to expand the function $\mu\to t_\mu^2/s_\mu(1^N)$ as a mixture of valuations of $s_\mu$. Are you aware of such representation? | |
| Jan 18, 2022 at 19:42 | history | asked | Marcel | CC BY-SA 4.0 |