CONE OF SPARSE f-VECTORS OF HALF-EULERIAN POSETS OF RANK 6


Extreme rays, given by interval systems:

   ( 1): empty family
   ( 2): [1,2]
   ( 3): [2,3]
   ( 4): [3,4]
   ( 5): [1,2][3,4]
   ( 6): [1,4]
   ( 7): [4,5]
   ( 8): [1,2][4,5]
   ( 9): [2,3][4,5]
   (10): [2,5]

Sparse flag f-vectors
   The entries f(S) in the sparse flag vector are represented by 
   variables xi as follows:

   x1  = f(emptyset)
   x2  = f(1)
   x3  = f(2)
   x4  = f(3)
   x5  = f(13)
   x6  = f(4)
   x7  = f(14)
   x8  = f(24)

Sparse flag f-vectors of extremes:

  ( 1) 1 1 1 1 1 1 1 1 
  ( 2) 0 1 1 0 1 0 1 1 
  ( 3) 0 0 1 1 1 0 0 1 
  ( 4) 0 0 0 1 1 1 1 1 
  ( 5) 0 0 0 0 1 0 1 1 
  ( 6) 0 1 1 1 1 1 1 1 
  ( 7) 0 0 0 0 0 1 1 1 
  ( 8) 0 0 0 0 0 0 1 1 
  ( 9) 0 0 0 0 0 0 0 1 
  (10) 0 0 1 1 1 1 1 1 

Inequalities for the cone:

  (F  1) -x1                      <= 0    
  (F  2)                   +x7-x8 <= 0    
  (F  3)          +x4-x5          <= 0    
  (F  4)       +x3   -x5          <= 0    
  (F  5)    +x2-x3                <= 0    
  (F  6) +x1-x2                   <= 0    
  (F  7)          -x4+x5+x6-x7    <= 0    
  (F  8)          +x4   -x6+x7-x8 <= 0    
  (F  9)    +x2-x3+x4   -x6       <= 0    
  (F 10) +x1-x2+x3-x4             <= 0    


Strong validity table.  A * in row i and column j indicates that 
extreme (j) lies on the facet given by inequality (F i).  The 
right-most column gives the number of extremes on each facet.
The bottom two rows, read vertically, give the number of facets
containing each extreme.

\ P      |             |
 \ O     |             |
I \ I    |             |
 N \ N   | 1     6     | #
  E \ T  |             |
   Q \ S |             |
    S \  |             |
       \ |             |
---------------------------
1        | .**** ***** :  9
2        | **.** ***.* :  8
3        | *.**. ***** :  8
4        | ***.. ***** :  8
5        | **.** ****. :  8
6        | *.*** .**** :  8
7        | ***** **.** :  9
8        | ***** *.*.* :  8
9        | ***** *.**. :  8
10       | ***.* .**** :  8
          .................
#        | 98888 88988