Grassmann.Betti

n=1:           n=2:
 \    ranks      \    ranks 
k \  0          k \  0   1 
   \----           \--------
 0 | 1           0 | 1  
 1 | 1           1 | 1   1  
                 2 | 1  

n=3:                  n=4: 
 \    ranks           \    ranks 
k \  0   1   2       k \  0   1   2   3   4 
   \------------        \--------------------
 0 | 1                 0 | 1  
 1 | 1   1   1         1 | 1   1   1   1  
 2 | 1   1   1         2 | 1   1   2   1   1  
 3 | 1                 3 | 1   1   1   1  
                       4 | 1  

n=5:                          n=6:  
 \    ranks                    \    ranks     
k \  0  1  2  3  4  5  6      k \  0  1  2  3  4  5  6  7  8  9 
   \----------------------       \------------------------------
 0 | 1                         0 | 1  
 1 | 1  1  1  1  1             1 | 1  1  1  1  1  1  
 2 | 1  1  2  2  2  1  1       2 | 1  1  2  2  3  2  2  1  1  
 3 | 1  1  2  2  2  1  1       3 | 1  1  2  3  3  3  3  2  1  1 
 4 | 1  1  1  1  1             4 | 1  1  2  2  3  2  2  1  1  
 5 | 1                         5 | 1  1  1  1  1  1  
                               6 | 1 

n=7:
 \    ranks 
k \  0  1  2  3  4  5  6  7  8  9 10 11 12 
   \----------------------------------------
 0 | 1  
 1 | 1  1  1  1  1  1  1  
 2 | 1  1  2  2  3  3  3  2  2  1  1  
 3 | 1  1  2  3  4  4  5  4  4  3  2  1  1  
 4 | 1  1  2  3  4  4  5  4  4  3  2  1  1  
 5 | 1  1  2  2  3  3  3  2  2  1  1  
 6 | 1  1  1  1  1  1  1  
 7 | 1  


n=8:
 \    ranks 
k \  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 
   \----------------------------------------------------
 0 | 1  
 1 | 1  1  1  1  1  1  1  1  
 2 | 1  1  2  2  3  3  4  3  3  2  2  1  1  
 3 | 1  1  2  3  4  5  6  6  6  6  5  4  3  2  1  1  
 4 | 1  1  2  3  5  5  7  7  8  7  7  5  5  3  2  1  1  
 5 | 1  1  2  3  4  5  6  6  6  6  5  4  3  2  1  1  
 6 | 1  1  2  2  3  3  4  3  3  2  2  1  1  
 7 | 1  1  1  1  1  1  1  1  
 8 | 1  


n=9:
 \    ranks 
k \  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 
   \-----------------------------------------------------------------
 0 | 1  
 1 | 1  1  1  1  1  1  1  1  1  
 2 | 1  1  2  2  3  3  4  4  4  3  3  2  2  1  1  
 3 | 1  1  2  3  4  5  7  7  8  8  8  7  7  5  4  3  2  1  1  
 4 | 1  1  2  3  5  6  8  9 11 11 12 11 11  9  8  6  5  3  2  1  1  
 5 | 1  1  2  3  5  6  8  9 11 11 12 11 11  9  8  6  5  3  2  1  1  
 6 | 1  1  2  3  4  5  7  7  8  8  8  7  7  5  4  3  2  1  1  
 7 | 1  1  2  2  3  3  4  4  4  3  3  2  2  1  1  
 8 | 1  1  1  1  1  1  1  1  1  
 9 | 1