Maximally inflected quintics
back to all Wronski dataFrequency vectors: degree followed by number of curves with the given number of isolated points isolated points.
| deg \ #i | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 4 | 0 | 0 | 3 | 2 | |||||||
| 5 | 0 | 0 | 0 | 12 | 18 | 9 | 3 | ||||
| 6 | 0 | 0 | 0 | 0 | 55 | 132 | 132 | 88 | 39 | 12 | 4 |
The minimal number of isolated points is $d-2$ by Kharlamov-Sottile, and the maximal number is the genus $\binom{d-1}{2}$.
Conjecture: The number of curves with minimal isolated points is the Fuß-Catalan number $\text{FC}_{3,d-2} = \frac{1}{2(d-2)+1}\binom{3(d-2)}{d-2}$, counting the number of $3$-ary trees with $d-2$ layers.
Conjecture: The number of curves with maximal isolated points is $d-2$. This is the orbit of the following tableau under promotion:
| $1$ | $2$ | ... | $d-2$ |
| $d-1$ | $d$ | ... | $2(d-2)$ |
| $2d-3$ | $2d-2$ | ... | $3(d-2)$ |
IP = number of isolated points
OS = orbit size (under promotion)
ED = number of extended descents
| Sottile index | Picture | Young tab | Isolated pts. | Promotion orbit size | Ext. descents | Chord diagram | notes | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 03 (praying mantis) |
 |
StandardTableau([[1,4,7],[2,5,8],[3,6,9]]) |
IP: 3 | OS: 3 | ED: 6 |  |
self-evacuative | |||||||||
| 39 (left squished praying mantis) |
 |
StandardTableau([[1,3,6],[2,4,7],[5,8,9]]) |
IP: 3 | OS: 3 | ED: 6 | |
||||||||||
| 40 (right squished praying mantis) |
 |
StandardTableau([[1,2,5],[3,6,8],[4,7,9]]) |
IP: 3 | OS: 3 | ED: 6 | |
||||||||||
| 06 (the mouse) |
 |
StandardTableau([[1,4,6],[2,5,7],[3,8,9]]) |
IP: 3 | OS: 9 | ED: 5 |  |
if we pick 1 to be the flex at $\infty$, we get a self-evacuative tableau | |||||||||
| 21 (asterisk) |
 |
StandardTableau([[1,3,5],[2,4,6],[7,8,9]]) |
IP: 3 | OS: 9 | ED: 5 | |
$p^8(\text{mouse})$ | |||||||||
| 22 |  |
StandardTableau([[1,2,3],[4,6,8],[5,7,9]]) |
IP: 3 | OS: 9 | ED: 5 | |
$p^2(\text{mouse})$ | |||||||||
| 41 (autograph) (and a closeup) |
 |
StandardTableau([[1,3,5],[2,4,6],[7,8,9]]) |
IP: 3 | OS: 9 | ED: 5 | |
$p^8(\text{mouse})$ | |||||||||
| 42 (lefty autograph) |
 |
StandardTableau([[1,2,7],[3,5,8],[4,6,9]]) |
IP: 3 | OS: 9 | ED: 5 | |
$p(\text{mouse})$ | |||||||||
| 07 (the co-R curve) |
 |
StandardTableau([[1,2,4],[3,5,8],[6,7,9]]) |
IP: 3 | OS: 9 | ED: 5 |  |
$p^7(\text{mouse})$ | |||||||||
| 08 (the R curve) |
 |
StandardTableau([[1,3,4],[2,5,7],[6,8,9]]) |
IP: 3 | Orbits size: 9 | ED: 5 | |
$p^3(\text{mouse})$ | |||||||||
| 33 (warrior I) |
 |
StandardTableau([[1,3,4],[2,5,7],[6,8,9]]) |
IP: 3 | OS: 9 | ED: 5 | |
$p^3(\text{mouse})$ | |||||||||
| 34 (warrior II) |
 |
StandardTableau([[1,2,4],[3,5,8],[6,7,9]]) |
IP: 3 | OS: 9 | ED: 5 | |
$p^7(\text{mouse})$ | |||||||||
| 02 (Ben Lomond) |
 |
StandardTableau([[1,2,4],[3,5,7],[6,8,9]]) |
IP: 4 | OS: 9 | ED: 5 |  |
if we pick 1 to be the flex at $\infty$, we get a self-evacuative tableau | |||||||||
| 13 (plough) |
 |
StandardTableau([[1,3,6],[2,4,8],[5,7,9]]) |
IP: 4 | OS: 9 | ED: 5 | |
in BL | |||||||||
| 14 |  |
StandardTableau([[1,3,5],[2,6,8],[4,7,9]]) |
IP: 4 | OS: 9 | ED: 5 | |
in BL | |||||||||
| 17 (matterhorn) |
 |
StandardTableau([[1,3,5],[2,4,7],[6,8,9]]) |
IP: 4 | OS: 9 | ED: 5 | |
In BL | |||||||||
| 18 (Kallur) |
 |
StandardTableau([[1,2,4],[3,6,8],[5,7,9]]) |
IP: 4 | OS: 9 | ED: 5 | |
in BL | |||||||||
| 27 |  |
StandardTableau([[1,3,5],[2,6,8],[4,7,9]]) |
IP: 4 | OS: 9 | ED: 5 | |
in BL | |||||||||
| 28 |  |
StandardTableau([[1,3,6],[2,4,8],[5,7,9]]) |
IP: 4 | OS: 9 | ED: 5 | |
in BL | |||||||||
| 35 (grebe) |
 |
StandardTableau([[1,3,6],[2,4,8],[5,7,9]]) |
IP: 4 | OS: 9 | ED: 5 | |
in BL | |||||||||
| 36 (co-grebe) |
 |
StandardTableau([[1,3,5],[2,6,8],[4,7,9]]) |
IP: 4 | OS: 9 | ED: 5 | |
in BL | |||||||||
| 01 (the mesa) |
 |
|
IP: 4 | OS: 9 | ED: 4 |  |
in sandcrawler | |||||||||
| 09 (sandcrawler) |
 |
|
IP: 4 | OS: 9 | ED: 4 | |
own orbit | |||||||||
| 10 (sandcrawler retreat) |
 |
|
IP: 4 | OS: 9 | ED: 4 | |
in sandcrawler | |||||||||
| 24 |  |
|
IP: 4 | |
||||||||||||
| 23 |  |
|
IP: 4 | |
Note: 23 and 24 should be vertically symmetric, so one of these pictures should be backwards |
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| 32 (zorro) |
 |
|
IP: 4 | |
||||||||||||
| 31 (co-zorro) |
 |
|
IP: 4 | |
||||||||||||
| 37 |  |
|
IP: 4 | |
||||||||||||
| 38 |  |
|
IP: 4 | |
||||||||||||
| 05 |  |
|
IP: 5 | |
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| 29 |  |
|
IP: 5 | |
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| 30 |  |
|
IP: 5 | |
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| 15 |  |
|
IP: 5 | |
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| 16 |  |
|
IP: 5 | |
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| 11 |  |
|
IP: 5 | |
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| 12 |  |
|
IP: 5 | |
||||||||||||
| 25 |  |
|
IP: 5 | |
||||||||||||
| 26 |  |
|
IP: 5 | OS: 9 | ED: 4 | |
||||||||||
| 04 (vendetta) |
 |
StandardTableau([[1,2,5],[3,4,8],[6,7,9]]) |
IP: 6 | OS: 3 | ED: 3 | |
||||||||||
| 19 |  |
StandardTableau([[1,3,4],[2,6,7],[5,8,9]]) |
IP: 6 | OS: 3 | ED: 3 | |
||||||||||
| 20 |  |
StandardTableau([[1,2,3],[4,5,6],[7,8,9]]) |
IP: 6 | OS: 3 | ED: 3 | |
