CONE OF FLAG l-VECTORS OF HALF-EULERIAN POSETS OF RANK 7 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] (11): [5,6] (12): [1,2][5,6] (13): [2,3][5,6] (14): [3,4][5,6] (15): [1,2][3,4][5,6] (16): [1,4][5,6] (17): [3,6] (18): [1,2][3,6] (19): [1,4][3,6] (20): [1,6] (21): "[1,2][2,6]+[2,5][5,6]" (22): "[1,3][3,4][4,6]+[1,2][2,3]+[4,5][5,6]" (23): "[1,2][3,4][4,5]+[3,5][5,6]+[1,2][2,5]" (24): "[1,2][2,4]+[2,3][3,4][5,6]+[2,5][5,6]" (Dual of (23)) Flag l-vectors The entries l(S) in the flag l-vector are represented by variables xi as follows: x1 = l(emptyset) x2 = l(12) x3 = l(23) x4 = l(34) x5 = l(1234) x6 = l(45) x7 = l(1245) x8 = l(2345) x9 = l(56) x10 = l(1256) x11 = l(2356) x12 = l(3456) x13 = l(123456) Flag l-vectors of extremes: ( 1) 1 0 0 0 0 0 0 0 0 0 0 0 0 ( 2) 1 -1 0 0 0 0 0 0 0 0 0 0 0 ( 3) 1 0 -1 0 0 0 0 0 0 0 0 0 0 ( 4) 1 0 0 -1 0 0 0 0 0 0 0 0 0 ( 5) 1 -1 0 -1 1 0 0 0 0 0 0 0 0 ( 6) 1 0 0 0 -1 0 0 0 0 0 0 0 0 ( 7) 1 0 0 0 0 -1 0 0 0 0 0 0 0 ( 8) 1 -1 0 0 0 -1 1 0 0 0 0 0 0 ( 9) 1 0 -1 0 0 -1 0 1 0 0 0 0 0 (10) 1 0 0 0 0 0 0 -1 0 0 0 0 0 (11) 1 0 0 0 0 0 0 0 -1 0 0 0 0 (12) 1 -1 0 0 0 0 0 0 -1 1 0 0 0 (13) 1 0 -1 0 0 0 0 0 -1 0 1 0 0 (14) 1 0 0 -1 0 0 0 0 -1 0 0 1 0 (15) 1 -1 0 -1 1 0 0 0 -1 1 0 1 -1 (16) 1 0 0 0 -1 0 0 0 -1 0 0 0 1 (17) 1 0 0 0 0 0 0 0 0 0 0 -1 0 (18) 1 -1 0 0 0 0 0 0 0 0 0 -1 1 (19) 1 0 0 0 -1 0 0 0 0 0 0 -1 1 (20) 1 0 0 0 0 0 0 0 0 0 0 0 -1 (21) 2 -1 0 0 0 0 0 -1 -1 0 0 0 1 (22) 3 -1 -1 -1 1 -1 0 0 -1 0 0 1 0 (23) 3 -2 0 -1 1 -1 1 -1 -1 0 0 1 0 (24) 3 -1 -1 -1 1 0 0 -1 -2 0 1 1 0 Inequalities for the cone: (L 1) +x4 <= 0 (L 2) +x6+x7 <= 0 (L 3) +x4 +x12 <= 0 (L 4) +x4+x5 <= 0 (L 5) +x3 +x11 <= 0 (L 6) +x9 +x11+x12 <= 0 (L 7) +x9+x10+x11 <= 0 (L 8) +x6+x7+x8 <= 0 (L 9) +x3 +x8 +x11 <= 0 (L 10) +x2 +x7 +x10 <= 0 (L 11) +x2 +x5 +x7 <= 0 (L 12) +x9+x10+x11+x12+x13 <= 0 (L 13) +x2 +x5 +x7 +x10 +x13 <= 0 (L 14) +x4+x5+x6+x7+x8 +x12+x13 <= 0 (L 15) +x3+x4+x5 +x8 +x11+x12+x13 <= 0 (L 16) -x7 <= 0 (L 17) -x10 <= 0 (L 18) -x11 <= 0 (L 19) +x4+x5+x6+x7+x8 -x10 +x12 <= 0 (L 20) +x3+x4+x5 +x8 -x10+x11+x12 <= 0 (L 21) +x4+x5+x6 -x10-x11+x12 <= 0 (L 22) +x3+x4+x5 -x7 -x10 +x12 <= 0 (L 23) -x1-x2-x3-x4-x5-x6-x7-x8-x9-x10-x11-x12-x13 <= 0 Strong validity table. A * in row i and column j indicates that extreme (j) lies on the facet given by inequality (L 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 11 16 21 | # E \ T | | Q \ S | | S \ | | \ | | -------------------------------------------- 1 | ***.. ***** ***.. ***** *... : 17 2 | ***** *.*.* ***** ***** *.** : 21 3 | ***.. ***** ***** *...* **** : 19 4 | ***.* .**** ***.* .**.* **** : 19 5 | **.** ***.* ***** ***** *.** : 21 6 | ***** ***** ..*** ....* .*** : 17 7 | ***** ***** .**.* .**** .... : 17 8 | ***** *.**. ***** ***** .... : 18 9 | **.** ****. ***** ***** .... : 18 10 | *.**. ***** ***** **.** .... : 17 11 | *.*** .**** *.*** .*..* .*** : 17 12 | ***** ***** .**** *.**. **** : 21 13 | *.*** .**** ***** ****. **** : 21 14 | ***.* ..**. ***** *.*.. **** : 17 15 | **..* .***. ***** *.*.. **** : 17 16 | ***** **.** ***** ***** **.* : 22 17 | ***** ***** *.**. ***** **** : 22 18 | ***** ***** **.** ***** ***. : 22 19 | ***.* ..**. *.*** ....* .*** : 14 20 | **..* .***. *.*** ....* .*** : 14 21 | ***.* ....* *..** ....* **** : 13 22 | **..* .*..* *..** ....* **** : 13 23 | .**** ***** ***** ***** **** : 23 .................................. # | 22112 11211 21222 11111 1111 | 20840 48097 06001 54539 5677