Timeline for number of weighted trivalent trees
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| Jun 6, 2011 at 8:07 | comment | added | F. C. | This Legendre transform, or rather the inversion formula obtained by derivating both sides, is related to the results and conjectures of fr.arxiv.org/abs/1010.3176 Indeed, the PreLie operad (dimension $n^{n-1}$ in degree $n$) contains a subspace of dimension $(n-2)!$ (related to the cyclic Lie operad), which conjecturally generates a free sub-operad (of dimension $(n-1)^{n-1}$ in degree $n$). | |
| Jun 5, 2011 at 19:49 | comment | added | Aleksey | This is indeed related to moduli spaces (of stable maps). The structure coefficients in formulas for genus 0 Gromov-Witten invariants are sums over $N$ marked trivalent trees. These can be bounded by the number of weighted trees in my question. I can show this number is bounded above by $C^N\cdot N!$, which is good enough for me. However, instead of adding another half a page proving this, I thought I might be able to quote something from the literature (and then apply Stirling's formula to get $C^N\cdot N!$). Since valence - 3 is so natural for trivalent, I thought this formula were known... | |
| Jun 5, 2011 at 17:39 | comment | added | F. C. | I do not know of representations of the symmetric group $S_n$ of dimension $(n-3)!$ or $(n-2)^(n-2)$ (I know only of representations of the symmetric group $S_n$ of dimension $(n-2)!$ or $(n-1)^(n-1)$). This would be needed to interpret this identity using the Koszul duality of cyclic operads. If you just want to prove it analytically, you should write these functions using $\log$ and the Lambert $W$-function. | |
| Jun 5, 2011 at 7:10 | history | edited | Dan Petersen | CC BY-SA 3.0 |
added 661 characters in body; added 1 characters in body
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| Jun 5, 2011 at 5:18 | history | answered | Dan Petersen | CC BY-SA 3.0 |