Timeline for Plane partitions not containing (1,1,1)
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Sep 14, 2020 at 21:47 | comment | added | Grigory M | @RichardStanley We also have some formulas for $\pi_{nm}=0$ in arXiv:1512.08779. In that case an answer is given by Brion's theorem. What really puzzles me is that the formula for $R(q)-1$ also looks like some version of Brion's theorem (each triple (x,y,z) in the answer below should correspond to a vertex etc)… | |
| Sep 14, 2020 at 18:13 | comment | added | Richard Stanley | Regard a plane partition as a two-dimensional array $(\pi_{ij})_{i,j\geq 1}$ in the usual way. The present question asks to count plane partition $n$ with $\pi_{22}\leq 2$. If $v(n)$ denotes the number of plane partitions of $n$ with $\pi_{22}=0$, then by EC1, Prop. 2.5.1, $\sum_{n\geq 0} v(n)q^n = \left( \sum_{m\geq 0} (-1)^mq^{{m+1\choose 2}}\right)\prod_{i\geq 1}(1-q^i)^{-2}$. This suggests counting plane partitions of $n$ with $\pi_{22}\leq k$ for $k\geq 2$ (and similar problems). | |
| Sep 13, 2020 at 20:11 | history | edited | Sam Hopkins |
edited tags
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| Apr 13, 2017 at 12:19 | history | edited | CommunityBot |
replaced http://math.stackexchange.com/ with https://math.stackexchange.com/
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| Jan 15, 2015 at 18:47 | vote | accept | Grigory M | ||
| Jan 13, 2015 at 13:12 | answer | added | Hjalmar Rosengren | timeline score: 8 | |
| Jan 9, 2015 at 10:43 | history | edited | Grigory M | CC BY-SA 3.0 |
typo in "context 2" fixed
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| Jan 9, 2015 at 10:39 | comment | added | Grigory M | @Hjalmar Yes, of course, thank you (I'll fix the typo) | |
| Jan 9, 2015 at 8:11 | comment | added | Hjalmar Rosengren | Am I correct that you mean $R(q)=1+\sum_{n=0}^\infty q^{n+1}\sum_{i+j+k=n}\genfrac[]{0pt}{}{i+j}i\genfrac[]{0pt}{}{j+k}k\genfrac[]{0pt}{}{k+i}k$ | |
| Jan 8, 2015 at 15:10 | history | asked | Grigory M | CC BY-SA 3.0 |