k=4

The cohomology of $C_2$-equivariant Grassmannians $\text{Gr}_k(\mathbb{R}^{p,q})$ when $k=4$

$k=4$	$q=1$	$q=2$	$q=3$	$q=4$	$q=5$
$p=8$	$\text{Gr}_4(\mathbb{R}^{8,1})$	6 possible	?	?
$p=9$	2 possible	?	?	?
$p=10$	16 possible	?	?	?	?
$p=11$	288 possible.	?	?	?	?

The cohomology of $\text{Gr}_4(\mathbb{R}^{8,1})$

Poincaré polynomial: $$x^{16} y^{4} + x^{15} y^{3} + 2 x^{14} y^{3} + 3 x^{13} y^{3} + 4 x^{12} y^{3} + x^{12} y^{2} + 3 x^{11} y^{3} + 2 x^{11} y^{2} + 2 x^{10} y^{3} + 5 x^{10} y^{2} + x^{9} y^{3} + 6 x^{9} y^{2} + 8 x^{8} y^{2} + 6 x^{7} y^{2} + x^{7} y + 5 x^{6} y^{2} + 2 x^{6} y + 2 x^{5} y^{2} + 3 x^{5} y + x^{4} y^{2} + 4 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ Generator count grid: Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_4(\mathbb{R}^{8,1}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2\oplus\Sigma^{2,1}\mathbb{M}_2\oplus\Sigma^{3,1}\mathbb{M}_2\oplus\Sigma^{4,1}\mathbb{M}_2\oplus\Sigma^{4,2}\mathbb{M}_2\oplus\Sigma^{5,1}\mathbb{M}_2\oplus\Sigma^{5,2}\mathbb{M}_2\oplus\Sigma^{6,1}\mathbb{M}_2\oplus\Sigma^{6,2}\mathbb{M}_2\oplus\Sigma^{7,1}\mathbb{M}_2\oplus\Sigma^{7,2}\mathbb{M}_2\oplus\Sigma^{8,2}\mathbb{M}_2\oplus\Sigma^{9,2}\mathbb{M}_2\oplus\Sigma^{9,3}\mathbb{M}_2\oplus\Sigma^{10,2}\mathbb{M}_2\oplus\Sigma^{10,3}\mathbb{M}_2\oplus\Sigma^{11,2}\mathbb{M}_2\oplus\Sigma^{11,3}\mathbb{M}_2\oplus\Sigma^{12,2}\mathbb{M}_2\oplus\Sigma^{12,3}\mathbb{M}_2\oplus\Sigma^{13,3}\mathbb{M}_2\oplus\Sigma^{14,3}\mathbb{M}_2\oplus\Sigma^{15,3}\mathbb{M}_2\oplus\Sigma^{16,4}\mathbb{M}_2.$$

The AutoKron can't determine the cohomology of $\text{Gr}_4(\mathbb{R}^{8,2})$.

There are 6 possibilities.
Here are their Poincaré polynomials: $$x^{16} y^{6} + x^{15} y^{6} + x^{14} y^{6} + x^{14} y^{5} + 3 x^{13} y^{5} + 4 x^{12} y^{5} + x^{12} y^{4} + 2 x^{11} y^{5} + 3 x^{11} y^{4} + 7 x^{10} y^{4} + 6 x^{9} y^{4} + x^{9} y^{3} + 5 x^{8} y^{4} + 3 x^{8} y^{3} + 2 x^{7} y^{4} + 4 x^{7} y^{3} + x^{7} y^{2} + 4 x^{6} y^{3} + 3 x^{6} y^{2} + 2 x^{5} y^{3} + 3 x^{5} y^{2} + 5 x^{4} y^{2} + 3 x^{3} y^{2} + x^{2} y^{2} + x^{2} y + x y + 1$$ $$x^{16} y^{6} + x^{15} y^{6} + x^{14} y^{6} + x^{14} y^{5} + 3 x^{13} y^{5} + 4 x^{12} y^{5} + x^{12} y^{4} + 2 x^{11} y^{5} + 3 x^{11} y^{4} + 7 x^{10} y^{4} + 6 x^{9} y^{4} + x^{9} y^{3} + 4 x^{8} y^{4} + 4 x^{8} y^{3} + 2 x^{7} y^{4} + 5 x^{7} y^{3} + 4 x^{6} y^{3} + 3 x^{6} y^{2} + 2 x^{5} y^{3} + 3 x^{5} y^{2} + 5 x^{4} y^{2} + 3 x^{3} y^{2} + x^{2} y^{2} + x^{2} y + x y + 1$$ $$x^{16} y^{6} + x^{15} y^{6} + x^{14} y^{6} + x^{14} y^{5} + 3 x^{13} y^{5} + 4 x^{12} y^{5} + x^{12} y^{4} + 2 x^{11} y^{5} + 3 x^{11} y^{4} + 7 x^{10} y^{4} + 6 x^{9} y^{4} + x^{9} y^{3} + 5 x^{8} y^{4} + 3 x^{8} y^{3} + x^{7} y^{4} + 5 x^{7} y^{3} + x^{7} y^{2} + 5 x^{6} y^{3} + 2 x^{6} y^{2} + 2 x^{5} y^{3} + 3 x^{5} y^{2} + 5 x^{4} y^{2} + 3 x^{3} y^{2} + x^{2} y^{2} + x^{2} y + x y + 1$$ $$x^{16} y^{6} + x^{15} y^{6} + x^{14} y^{6} + x^{14} y^{5} + 3 x^{13} y^{5} + 4 x^{12} y^{5} + x^{12} y^{4} + 2 x^{11} y^{5} + 3 x^{11} y^{4} + 7 x^{10} y^{4} + 6 x^{9} y^{4} + x^{9} y^{3} + 4 x^{8} y^{4} + 4 x^{8} y^{3} + x^{7} y^{4} + 6 x^{7} y^{3} + 5 x^{6} y^{3} + 2 x^{6} y^{2} + 2 x^{5} y^{3} + 3 x^{5} y^{2} + 5 x^{4} y^{2} + 3 x^{3} y^{2} + x^{2} y^{2} + x^{2} y + x y + 1$$ $$x^{16} y^{6} + x^{15} y^{6} + x^{14} y^{6} + x^{14} y^{5} + 3 x^{13} y^{5} + 4 x^{12} y^{5} + x^{12} y^{4} + 2 x^{11} y^{5} + 3 x^{11} y^{4} + 7 x^{10} y^{4} + 6 x^{9} y^{4} + x^{9} y^{3} + 5 x^{8} y^{4} + 3 x^{8} y^{3} + 6 x^{7} y^{3} + x^{7} y^{2} + 6 x^{6} y^{3} + x^{6} y^{2} + 2 x^{5} y^{3} + 3 x^{5} y^{2} + 5 x^{4} y^{2} + 3 x^{3} y^{2} + x^{2} y^{2} + x^{2} y + x y + 1$$ $$x^{16} y^{6} + x^{15} y^{6} + x^{14} y^{6} + x^{14} y^{5} + 3 x^{13} y^{5} + 4 x^{12} y^{5} + x^{12} y^{4} + 2 x^{11} y^{5} + 3 x^{11} y^{4} + 7 x^{10} y^{4} + 6 x^{9} y^{4} + x^{9} y^{3} + 4 x^{8} y^{4} + 4 x^{8} y^{3} + 7 x^{7} y^{3} + 6 x^{6} y^{3} + x^{6} y^{2} + 2 x^{5} y^{3} + 3 x^{5} y^{2} + 5 x^{4} y^{2} + 3 x^{3} y^{2} + x^{2} y^{2} + x^{2} y + x y + 1$$ Here are the corresponding generator grids to these 6 possibilities:

The AutoKron can't determine the cohomology of $\text{Gr}_4(\mathbb{R}^{9,1})$.

There are 2 possibilities.
Here are their Poincaré polynomials: $$x^{20} y^{4} + x^{19} y^{4} + x^{18} y^{4} + x^{18} y^{3} + x^{17} y^{4} + 2 x^{17} y^{3} + x^{16} y^{4} + 4 x^{16} y^{3} + 6 x^{15} y^{3} + 7 x^{14} y^{3} + x^{14} y^{2} + 7 x^{13} y^{3} + 2 x^{13} y^{2} + 6 x^{12} y^{3} + 5 x^{12} y^{2} + 4 x^{11} y^{3} + 7 x^{11} y^{2} + 2 x^{10} y^{3} + 10 x^{10} y^{2} + x^{9} y^{3} + 10 x^{9} y^{2} + 10 x^{8} y^{2} + x^{8} y + 7 x^{7} y^{2} + 2 x^{7} y + 5 x^{6} y^{2} + 3 x^{6} y + 2 x^{5} y^{2} + 4 x^{5} y + x^{4} y^{2} + 4 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ $$x^{20} y^{4} + x^{19} y^{4} + x^{18} y^{4} + x^{18} y^{3} + x^{17} y^{4} + 2 x^{17} y^{3} + x^{16} y^{4} + 4 x^{16} y^{3} + 6 x^{15} y^{3} + 7 x^{14} y^{3} + x^{14} y^{2} + 7 x^{13} y^{3} + 2 x^{13} y^{2} + 6 x^{12} y^{3} + 5 x^{12} y^{2} + 4 x^{11} y^{3} + 7 x^{11} y^{2} + 2 x^{10} y^{3} + 10 x^{10} y^{2} + 11 x^{9} y^{2} + 11 x^{8} y^{2} + 7 x^{7} y^{2} + 2 x^{7} y + 5 x^{6} y^{2} + 3 x^{6} y + 2 x^{5} y^{2} + 4 x^{5} y + x^{4} y^{2} + 4 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ Here are the corresponding generator grids to these 2 possibilities:

The AutoKron can't determine the cohomology of $\text{Gr}_4(\mathbb{R}^{10,1})$.

There are 16 possibilities.
Here are their Poincaré polynomials: $$x^{24} y^{4} + x^{23} y^{4} + 2 x^{22} y^{4} + 2 x^{21} y^{4} + x^{21} y^{3} + 3 x^{20} y^{4} + 2 x^{20} y^{3} + 2 x^{19} y^{4} + 4 x^{19} y^{3} + 2 x^{18} y^{4} + 7 x^{18} y^{3} + x^{17} y^{4} + 9 x^{17} y^{3} + x^{16} y^{4} + 11 x^{16} y^{3} + x^{16} y^{2} + 12 x^{15} y^{3} + 2 x^{15} y^{2} + 11 x^{14} y^{3} + 5 x^{14} y^{2} + 9 x^{13} y^{3} + 7 x^{13} y^{2} + 7 x^{12} y^{3} + 11 x^{12} y^{2} + 4 x^{11} y^{3} + 12 x^{11} y^{2} + 2 x^{10} y^{3} + 14 x^{10} y^{2} + x^{9} y^{3} + 12 x^{9} y^{2} + x^{9} y + 11 x^{8} y^{2} + 2 x^{8} y + 7 x^{7} y^{2} + 3 x^{7} y + 5 x^{6} y^{2} + 4 x^{6} y + 2 x^{5} y^{2} + 4 x^{5} y + x^{4} y^{2} + 4 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ $$x^{24} y^{4} + x^{23} y^{4} + 2 x^{22} y^{4} + 2 x^{21} y^{4} + x^{21} y^{3} + 3 x^{20} y^{4} + 2 x^{20} y^{3} + 2 x^{19} y^{4} + 4 x^{19} y^{3} + 2 x^{18} y^{4} + 7 x^{18} y^{3} + 10 x^{17} y^{3} + x^{16} y^{4} + 12 x^{16} y^{3} + 12 x^{15} y^{3} + 2 x^{15} y^{2} + 11 x^{14} y^{3} + 5 x^{14} y^{2} + 9 x^{13} y^{3} + 7 x^{13} y^{2} + 7 x^{12} y^{3} + 11 x^{12} y^{2} + 4 x^{11} y^{3} + 12 x^{11} y^{2} + 2 x^{10} y^{3} + 14 x^{10} y^{2} + x^{9} y^{3} + 12 x^{9} y^{2} + x^{9} y + 11 x^{8} y^{2} + 2 x^{8} y + 7 x^{7} y^{2} + 3 x^{7} y + 5 x^{6} y^{2} + 4 x^{6} y + 2 x^{5} y^{2} + 4 x^{5} y + x^{4} y^{2} + 4 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ $$x^{24} y^{4} + x^{23} y^{4} + 2 x^{22} y^{4} + 2 x^{21} y^{4} + x^{21} y^{3} + 3 x^{20} y^{4} + 2 x^{20} y^{3} + 2 x^{19} y^{4} + 4 x^{19} y^{3} + 2 x^{18} y^{4} + 7 x^{18} y^{3} + x^{17} y^{4} + 9 x^{17} y^{3} + 12 x^{16} y^{3} + x^{16} y^{2} + 13 x^{15} y^{3} + x^{15} y^{2} + 11 x^{14} y^{3} + 5 x^{14} y^{2} + 9 x^{13} y^{3} + 7 x^{13} y^{2} + 7 x^{12} y^{3} + 11 x^{12} y^{2} + 4 x^{11} y^{3} + 12 x^{11} y^{2} + 2 x^{10} y^{3} + 14 x^{10} y^{2} + x^{9} y^{3} + 12 x^{9} y^{2} + x^{9} y + 11 x^{8} y^{2} + 2 x^{8} y + 7 x^{7} y^{2} + 3 x^{7} y + 5 x^{6} y^{2} + 4 x^{6} y + 2 x^{5} y^{2} + 4 x^{5} y + x^{4} y^{2} + 4 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ $$x^{24} y^{4} + x^{23} y^{4} + 2 x^{22} y^{4} + 2 x^{21} y^{4} + x^{21} y^{3} + 3 x^{20} y^{4} + 2 x^{20} y^{3} + 2 x^{19} y^{4} + 4 x^{19} y^{3} + 2 x^{18} y^{4} + 7 x^{18} y^{3} + 10 x^{17} y^{3} + 13 x^{16} y^{3} + 13 x^{15} y^{3} + x^{15} y^{2} + 11 x^{14} y^{3} + 5 x^{14} y^{2} + 9 x^{13} y^{3} + 7 x^{13} y^{2} + 7 x^{12} y^{3} + 11 x^{12} y^{2} + 4 x^{11} y^{3} + 12 x^{11} y^{2} + 2 x^{10} y^{3} + 14 x^{10} y^{2} + x^{9} y^{3} + 12 x^{9} y^{2} + x^{9} y + 11 x^{8} y^{2} + 2 x^{8} y + 7 x^{7} y^{2} + 3 x^{7} y + 5 x^{6} y^{2} + 4 x^{6} y + 2 x^{5} y^{2} + 4 x^{5} y + x^{4} y^{2} + 4 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ $$x^{24} y^{4} + x^{23} y^{4} + 2 x^{22} y^{4} + 2 x^{21} y^{4} + x^{21} y^{3} + 3 x^{20} y^{4} + 2 x^{20} y^{3} + 2 x^{19} y^{4} + 4 x^{19} y^{3} + 2 x^{18} y^{4} + 7 x^{18} y^{3} + x^{17} y^{4} + 9 x^{17} y^{3} + x^{16} y^{4} + 11 x^{16} y^{3} + x^{16} y^{2} + 12 x^{15} y^{3} + 2 x^{15} y^{2} + 11 x^{14} y^{3} + 5 x^{14} y^{2} + 9 x^{13} y^{3} + 7 x^{13} y^{2} + 7 x^{12} y^{3} + 11 x^{12} y^{2} + 4 x^{11} y^{3} + 12 x^{11} y^{2} + x^{10} y^{3} + 15 x^{10} y^{2} + x^{9} y^{3} + 13 x^{9} y^{2} + 11 x^{8} y^{2} + 2 x^{8} y + 7 x^{7} y^{2} + 3 x^{7} y + 5 x^{6} y^{2} + 4 x^{6} y + 2 x^{5} y^{2} + 4 x^{5} y + x^{4} y^{2} + 4 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ $$x^{24} y^{4} + x^{23} y^{4} + 2 x^{22} y^{4} + 2 x^{21} y^{4} + x^{21} y^{3} + 3 x^{20} y^{4} + 2 x^{20} y^{3} + 2 x^{19} y^{4} + 4 x^{19} y^{3} + 2 x^{18} y^{4} + 7 x^{18} y^{3} + 10 x^{17} y^{3} + x^{16} y^{4} + 12 x^{16} y^{3} + 12 x^{15} y^{3} + 2 x^{15} y^{2} + 11 x^{14} y^{3} + 5 x^{14} y^{2} + 9 x^{13} y^{3} + 7 x^{13} y^{2} + 7 x^{12} y^{3} + 11 x^{12} y^{2} + 4 x^{11} y^{3} + 12 x^{11} y^{2} + x^{10} y^{3} + 15 x^{10} y^{2} + x^{9} y^{3} + 13 x^{9} y^{2} + 11 x^{8} y^{2} + 2 x^{8} y + 7 x^{7} y^{2} + 3 x^{7} y + 5 x^{6} y^{2} + 4 x^{6} y + 2 x^{5} y^{2} + 4 x^{5} y + x^{4} y^{2} + 4 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ $$x^{24} y^{4} + x^{23} y^{4} + 2 x^{22} y^{4} + 2 x^{21} y^{4} + x^{21} y^{3} + 3 x^{20} y^{4} + 2 x^{20} y^{3} + 2 x^{19} y^{4} + 4 x^{19} y^{3} + 2 x^{18} y^{4} + 7 x^{18} y^{3} + x^{17} y^{4} + 9 x^{17} y^{3} + 12 x^{16} y^{3} + x^{16} y^{2} + 13 x^{15} y^{3} + x^{15} y^{2} + 11 x^{14} y^{3} + 5 x^{14} y^{2} + 9 x^{13} y^{3} + 7 x^{13} y^{2} + 7 x^{12} y^{3} + 11 x^{12} y^{2} + 4 x^{11} y^{3} + 12 x^{11} y^{2} + x^{10} y^{3} + 15 x^{10} y^{2} + x^{9} y^{3} + 13 x^{9} y^{2} + 11 x^{8} y^{2} + 2 x^{8} y + 7 x^{7} y^{2} + 3 x^{7} y + 5 x^{6} y^{2} + 4 x^{6} y + 2 x^{5} y^{2} + 4 x^{5} y + x^{4} y^{2} + 4 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ $$x^{24} y^{4} + x^{23} y^{4} + 2 x^{22} y^{4} + 2 x^{21} y^{4} + x^{21} y^{3} + 3 x^{20} y^{4} + 2 x^{20} y^{3} + 2 x^{19} y^{4} + 4 x^{19} y^{3} + 2 x^{18} y^{4} + 7 x^{18} y^{3} + 10 x^{17} y^{3} + 13 x^{16} y^{3} + 13 x^{15} y^{3} + x^{15} y^{2} + 11 x^{14} y^{3} + 5 x^{14} y^{2} + 9 x^{13} y^{3} + 7 x^{13} y^{2} + 7 x^{12} y^{3} + 11 x^{12} y^{2} + 4 x^{11} y^{3} + 12 x^{11} y^{2} + x^{10} y^{3} + 15 x^{10} y^{2} + x^{9} y^{3} + 13 x^{9} y^{2} + 11 x^{8} y^{2} + 2 x^{8} y + 7 x^{7} y^{2} + 3 x^{7} y + 5 x^{6} y^{2} + 4 x^{6} y + 2 x^{5} y^{2} + 4 x^{5} y + x^{4} y^{2} + 4 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ $$x^{24} y^{4} + x^{23} y^{4} + 2 x^{22} y^{4} + 2 x^{21} y^{4} + x^{21} y^{3} + 3 x^{20} y^{4} + 2 x^{20} y^{3} + 2 x^{19} y^{4} + 4 x^{19} y^{3} + 2 x^{18} y^{4} + 7 x^{18} y^{3} + x^{17} y^{4} + 9 x^{17} y^{3} + x^{16} y^{4} + 11 x^{16} y^{3} + x^{16} y^{2} + 12 x^{15} y^{3} + 2 x^{15} y^{2} + 11 x^{14} y^{3} + 5 x^{14} y^{2} + 9 x^{13} y^{3} + 7 x^{13} y^{2} + 7 x^{12} y^{3} + 11 x^{12} y^{2} + 4 x^{11} y^{3} + 12 x^{11} y^{2} + 2 x^{10} y^{3} + 14 x^{10} y^{2} + 13 x^{9} y^{2} + x^{9} y + 12 x^{8} y^{2} + x^{8} y + 7 x^{7} y^{2} + 3 x^{7} y + 5 x^{6} y^{2} + 4 x^{6} y + 2 x^{5} y^{2} + 4 x^{5} y + x^{4} y^{2} + 4 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ $$x^{24} y^{4} + x^{23} y^{4} + 2 x^{22} y^{4} + 2 x^{21} y^{4} + x^{21} y^{3} + 3 x^{20} y^{4} + 2 x^{20} y^{3} + 2 x^{19} y^{4} + 4 x^{19} y^{3} + 2 x^{18} y^{4} + 7 x^{18} y^{3} + 10 x^{17} y^{3} + x^{16} y^{4} + 12 x^{16} y^{3} + 12 x^{15} y^{3} + 2 x^{15} y^{2} + 11 x^{14} y^{3} + 5 x^{14} y^{2} + 9 x^{13} y^{3} + 7 x^{13} y^{2} + 7 x^{12} y^{3} + 11 x^{12} y^{2} + 4 x^{11} y^{3} + 12 x^{11} y^{2} + 2 x^{10} y^{3} + 14 x^{10} y^{2} + 13 x^{9} y^{2} + x^{9} y + 12 x^{8} y^{2} + x^{8} y + 7 x^{7} y^{2} + 3 x^{7} y + 5 x^{6} y^{2} + 4 x^{6} y + 2 x^{5} y^{2} + 4 x^{5} y + x^{4} y^{2} + 4 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ $$x^{24} y^{4} + x^{23} y^{4} + 2 x^{22} y^{4} + 2 x^{21} y^{4} + x^{21} y^{3} + 3 x^{20} y^{4} + 2 x^{20} y^{3} + 2 x^{19} y^{4} + 4 x^{19} y^{3} + 2 x^{18} y^{4} + 7 x^{18} y^{3} + x^{17} y^{4} + 9 x^{17} y^{3} + 12 x^{16} y^{3} + x^{16} y^{2} + 13 x^{15} y^{3} + x^{15} y^{2} + 11 x^{14} y^{3} + 5 x^{14} y^{2} + 9 x^{13} y^{3} + 7 x^{13} y^{2} + 7 x^{12} y^{3} + 11 x^{12} y^{2} + 4 x^{11} y^{3} + 12 x^{11} y^{2} + 2 x^{10} y^{3} + 14 x^{10} y^{2} + 13 x^{9} y^{2} + x^{9} y + 12 x^{8} y^{2} + x^{8} y + 7 x^{7} y^{2} + 3 x^{7} y + 5 x^{6} y^{2} + 4 x^{6} y + 2 x^{5} y^{2} + 4 x^{5} y + x^{4} y^{2} + 4 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ $$x^{24} y^{4} + x^{23} y^{4} + 2 x^{22} y^{4} + 2 x^{21} y^{4} + x^{21} y^{3} + 3 x^{20} y^{4} + 2 x^{20} y^{3} + 2 x^{19} y^{4} + 4 x^{19} y^{3} + 2 x^{18} y^{4} + 7 x^{18} y^{3} + 10 x^{17} y^{3} + 13 x^{16} y^{3} + 13 x^{15} y^{3} + x^{15} y^{2} + 11 x^{14} y^{3} + 5 x^{14} y^{2} + 9 x^{13} y^{3} + 7 x^{13} y^{2} + 7 x^{12} y^{3} + 11 x^{12} y^{2} + 4 x^{11} y^{3} + 12 x^{11} y^{2} + 2 x^{10} y^{3} + 14 x^{10} y^{2} + 13 x^{9} y^{2} + x^{9} y + 12 x^{8} y^{2} + x^{8} y + 7 x^{7} y^{2} + 3 x^{7} y + 5 x^{6} y^{2} + 4 x^{6} y + 2 x^{5} y^{2} + 4 x^{5} y + x^{4} y^{2} + 4 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ $$x^{24} y^{4} + x^{23} y^{4} + 2 x^{22} y^{4} + 2 x^{21} y^{4} + x^{21} y^{3} + 3 x^{20} y^{4} + 2 x^{20} y^{3} + 2 x^{19} y^{4} + 4 x^{19} y^{3} + 2 x^{18} y^{4} + 7 x^{18} y^{3} + x^{17} y^{4} + 9 x^{17} y^{3} + x^{16} y^{4} + 11 x^{16} y^{3} + x^{16} y^{2} + 12 x^{15} y^{3} + 2 x^{15} y^{2} + 11 x^{14} y^{3} + 5 x^{14} y^{2} + 9 x^{13} y^{3} + 7 x^{13} y^{2} + 7 x^{12} y^{3} + 11 x^{12} y^{2} + 4 x^{11} y^{3} + 12 x^{11} y^{2} + x^{10} y^{3} + 15 x^{10} y^{2} + 14 x^{9} y^{2} + 12 x^{8} y^{2} + x^{8} y + 7 x^{7} y^{2} + 3 x^{7} y + 5 x^{6} y^{2} + 4 x^{6} y + 2 x^{5} y^{2} + 4 x^{5} y + x^{4} y^{2} + 4 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ $$x^{24} y^{4} + x^{23} y^{4} + 2 x^{22} y^{4} + 2 x^{21} y^{4} + x^{21} y^{3} + 3 x^{20} y^{4} + 2 x^{20} y^{3} + 2 x^{19} y^{4} + 4 x^{19} y^{3} + 2 x^{18} y^{4} + 7 x^{18} y^{3} + 10 x^{17} y^{3} + x^{16} y^{4} + 12 x^{16} y^{3} + 12 x^{15} y^{3} + 2 x^{15} y^{2} + 11 x^{14} y^{3} + 5 x^{14} y^{2} + 9 x^{13} y^{3} + 7 x^{13} y^{2} + 7 x^{12} y^{3} + 11 x^{12} y^{2} + 4 x^{11} y^{3} + 12 x^{11} y^{2} + x^{10} y^{3} + 15 x^{10} y^{2} + 14 x^{9} y^{2} + 12 x^{8} y^{2} + x^{8} y + 7 x^{7} y^{2} + 3 x^{7} y + 5 x^{6} y^{2} + 4 x^{6} y + 2 x^{5} y^{2} + 4 x^{5} y + x^{4} y^{2} + 4 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ $$x^{24} y^{4} + x^{23} y^{4} + 2 x^{22} y^{4} + 2 x^{21} y^{4} + x^{21} y^{3} + 3 x^{20} y^{4} + 2 x^{20} y^{3} + 2 x^{19} y^{4} + 4 x^{19} y^{3} + 2 x^{18} y^{4} + 7 x^{18} y^{3} + x^{17} y^{4} + 9 x^{17} y^{3} + 12 x^{16} y^{3} + x^{16} y^{2} + 13 x^{15} y^{3} + x^{15} y^{2} + 11 x^{14} y^{3} + 5 x^{14} y^{2} + 9 x^{13} y^{3} + 7 x^{13} y^{2} + 7 x^{12} y^{3} + 11 x^{12} y^{2} + 4 x^{11} y^{3} + 12 x^{11} y^{2} + x^{10} y^{3} + 15 x^{10} y^{2} + 14 x^{9} y^{2} + 12 x^{8} y^{2} + x^{8} y + 7 x^{7} y^{2} + 3 x^{7} y + 5 x^{6} y^{2} + 4 x^{6} y + 2 x^{5} y^{2} + 4 x^{5} y + x^{4} y^{2} + 4 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ $$x^{24} y^{4} + x^{23} y^{4} + 2 x^{22} y^{4} + 2 x^{21} y^{4} + x^{21} y^{3} + 3 x^{20} y^{4} + 2 x^{20} y^{3} + 2 x^{19} y^{4} + 4 x^{19} y^{3} + 2 x^{18} y^{4} + 7 x^{18} y^{3} + 10 x^{17} y^{3} + 13 x^{16} y^{3} + 13 x^{15} y^{3} + x^{15} y^{2} + 11 x^{14} y^{3} + 5 x^{14} y^{2} + 9 x^{13} y^{3} + 7 x^{13} y^{2} + 7 x^{12} y^{3} + 11 x^{12} y^{2} + 4 x^{11} y^{3} + 12 x^{11} y^{2} + x^{10} y^{3} + 15 x^{10} y^{2} + 14 x^{9} y^{2} + 12 x^{8} y^{2} + x^{8} y + 7 x^{7} y^{2} + 3 x^{7} y + 5 x^{6} y^{2} + 4 x^{6} y + 2 x^{5} y^{2} + 4 x^{5} y + x^{4} y^{2} + 4 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ Here are the corresponding generator grids to these 16 possibilities: