CONE OF SPARSE f-VECTORS OF HALF-EULERIAN POSETS OF RANK 5


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]

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)

Sparse flag f-vectors of extremes:

  ( 1) 1 1 1 1 1 
  ( 2) 0 1 1 0 1 
  ( 3) 0 0 1 1 1 
  ( 4) 0 0 0 1 1 
  ( 5) 0 0 0 0 1 
  ( 6) 0 1 1 1 1 

Inequalities for the cone:

  (F  1) -x1             <= 0    
  (F  2)          +x4-x5 <= 0    
  (F  3)       +x3   -x5 <= 0    
  (F  4)    +x2-x3       <= 0    
  (F  5) +x1-x2          <= 0    
  (F  6) +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        | .**** * :  5
2        | *.**. * :  4
3        | ***.. * :  4
4        | **.** * :  5
5        | *.*** . :  4
6        | ***.* . :  4
          .............
#        | 54544 4