Timeline for Inequality for hook numbers in Young diagrams
Current License: CC BY-SA 4.0
20 events
| when toggle format | what | by | license | comment | |
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| Jan 31 at 9:02 | answer | added | Bruno Le Floch | timeline score: 0 | |
| Jul 24, 2020 at 21:13 | answer | added | Gjergji Zaimi | timeline score: 5 | |
| Mar 31, 2019 at 21:43 | comment | added | Igor Pak | One more thing. This is all very mysterious until you notice the similarities of the proof with the incremental proofs of isoperimetric inequalities. Say, you are trying to prove that the square is optimal in $(\mathbb R^2, \ell_\infty)$. What do you do? | |
| Mar 31, 2019 at 21:34 | comment | added | Igor Pak | @GjergjiZaimi Right. Since the squares in the proof are sliding only along the axis, there indeed might be a partially commutative and perhaps q-commutative version, in the sense of Carier-Foata (see e.g. math.ucla.edu/~pak/papers/qmm7.pdf for defs). I am not sure if that's what you meant. | |
| Mar 31, 2019 at 18:05 | comment | added | Gjergji Zaimi | I am fascinated by this phenomenon and I'm glad you figured out he case of trees as well. For a general poset $P$ we can define the hook lengths as the exponents in the denominator of the generating function of its P-partitions, and the contents as the same but for the dual poset. It seems like hooks will majorize contents whenever $P$ is a D-complete poset (and maybe more), and this seems to signal that this story might also have a commutative algebra analog. | |
| Mar 31, 2019 at 1:00 | history | edited | Igor Pak | CC BY-SA 4.0 |
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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 8, 2017 at 19:10 | comment | added | Gjergji Zaimi | It seems to me like the majorization inequality should hold for trees as well (and maybe other classes of posets) but I haven't been able to adapt Fedor's proof. As far as the weaker statement that the moments of the sequence of hooks are greater than the moments of the complementary hooks, that has a simple combinatorial proof for young diagrams and trees alike. | |
| Jul 10, 2016 at 0:00 | history | edited | Igor Pak | CC BY-SA 3.0 |
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| Jul 9, 2016 at 0:48 | history | edited | Igor Pak | CC BY-SA 3.0 |
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| Jul 7, 2016 at 23:49 | vote | accept | Igor Pak | ||
| Jul 7, 2016 at 20:56 | answer | added | Fedor Petrov | timeline score: 18 | |
| Jul 7, 2016 at 6:17 | comment | added | Fedor Petrov | $k$ squares with minimal complimentary hooks form a Young subdiagram of $\lambda$ situated in first few diagonals (we may choose them so, at least). and what we want is to find $k$ squares in $\lambda$ with at most as large sum of hooks. Maybe, some clever rearrangements work. | |
| Jul 7, 2016 at 5:48 | comment | added | Igor Pak | @Fёdor: This is a nice idea. While I don't have a counterexample, there is no natural ordering on squares of $\lambda$, so I sort of doubt that. | |
| Jul 7, 2016 at 4:43 | history | edited | Richard Stanley | CC BY-SA 3.0 |
changed "complimentary" to "complementary"
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| Jul 7, 2016 at 4:40 | comment | added | Fedor Petrov | Is it true that the array $(h)$ majorates $(q)$, then the inequality would follow from Karamata inequality for the concave function $\log$? | |
| Jul 7, 2016 at 4:23 | history | edited | Igor Pak | CC BY-SA 3.0 |
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| Jul 7, 2016 at 3:18 | history | edited | Igor Pak | CC BY-SA 3.0 |
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| Jul 7, 2016 at 2:00 | history | edited | Igor Pak | CC BY-SA 3.0 |
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| Jul 7, 2016 at 1:53 | history | asked | Igor Pak | CC BY-SA 3.0 |