CONE OF SPARSE f-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)) 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) x9 = f(5) x10 = f(15) x11 = f(25) x12 = f(35) x13 = f(135) Sparse flag f-vectors of extremes: ( 1) 1 1 1 1 1 1 1 1 1 1 1 1 1 ( 2) 0 1 1 0 1 0 1 1 0 1 1 0 1 ( 3) 0 0 1 1 1 0 0 1 0 0 1 1 1 ( 4) 0 0 0 1 1 1 1 1 0 0 0 1 1 ( 5) 0 0 0 0 1 0 1 1 0 0 0 0 1 ( 6) 0 1 1 1 1 1 1 1 0 1 1 1 1 ( 7) 0 0 0 0 0 1 1 1 1 1 1 1 1 ( 8) 0 0 0 0 0 0 1 1 0 1 1 0 1 ( 9) 0 0 0 0 0 0 0 1 0 0 1 1 1 (10) 0 0 1 1 1 1 1 1 1 1 1 1 1 (11) 0 0 0 0 0 0 0 0 1 1 1 1 1 (12) 0 0 0 0 0 0 0 0 0 1 1 0 1 (13) 0 0 0 0 0 0 0 0 0 0 1 1 1 (14) 0 0 0 0 0 0 0 0 0 0 0 1 1 (15) 0 0 0 0 0 0 0 0 0 0 0 0 1 (16) 0 0 0 0 0 0 0 0 0 1 1 1 1 (17) 0 0 0 1 1 1 1 1 1 1 1 1 1 (18) 0 0 0 0 1 0 1 1 0 1 1 0 1 (19) 0 0 0 1 1 1 1 1 0 1 1 1 1 (20) 0 1 1 1 1 1 1 1 1 1 1 1 1 (21) 0 0 1 0 1 0 1 1 1 2 2 1 2 (22) 0 0 1 0 1 0 1 2 1 1 2 2 3 (23) 0 0 1 0 1 0 2 2 1 2 2 1 3 (24) 0 0 1 0 1 0 1 1 1 1 2 2 3 Inequalities for the cone: (F 1) -x1 <= 0 (F 2) +x11 -x13 <= 0 (F 3) +x7-x8 <= 0 (F 4) +x4-x5 <= 0 (F 5) +x3 -x5 <= 0 (F 6) +x2-x3 <= 0 (F 7) +x1-x2 <= 0 (F 8) +x1 -x4 <= 0 (F 9) +x1 - x6 <= 0 (F 10) -x4+x5+ x6-x7 <= 0 (F 11) - x6+x7 +x12-x13 <= 0 (F 12) -x7+x8 +x10-x11 <= 0 (F 13) +x9-x10+x11-x12 <= 0 (F 14) + x6 -x10+x11-x12 <= 0 (F 15) +x4 - x6+x7-x8 <= 0 (F 16) +x3-x4 +x9 -x11 <= 0 (F 17) +x3 -x5 +x7 -x10 <= 0 (F 18) +x3 - x6 +x9 -x11 <= 0 (F 19) +x2-x3+x4 - x6 <= 0 (F 20) +x4-x5- x6+x7 +x9-x10 <= 0 (F 21) +x2 -x5- x6+x7 +x9-x10 <= 0 (F 22) +x1-x2+x3-x4 + x6 -x9 <= 0 (F 23) +x2 +x4-x5-2x6+x7 +x9-x10 <= 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 11 16 21 | # E \ T | | Q \ S | | S \ | | \ | | -------------------------------------------- 1 | .**** ***** ***** ***** **** : 23 2 | ***.. ***** ***.. ***** *... : 17 3 | **.** ***.* ***** ***** *.** : 21 4 | *.**. ***** ***** **.** .... : 17 5 | ***.. ***** ***** *...* **** : 19 6 | **.** ****. ***** ***** .... : 18 7 | *.*** .**** ***** ****. **** : 21 8 | **..* .***. ***** *.*.. **** : 17 9 | ***.* ..**. ***** *.*.. **** : 17 10 | ***** **.** ***** ***** **.* : 22 11 | ***** ***** *.**. ***** **** : 22 12 | ***** ***** **.** ***** ***. : 22 13 | ***.* .**** ***.* .**.* **** : 19 14 | ***** ***** .**.* .**** .... : 17 15 | ***** *.*.* ***** ***** *.** : 21 16 | **..* .*..* *..** ....* **** : 13 17 | ***** ***** ..*** ....* .*** : 17 18 | ***.* ....* *..** ....* **** : 13 19 | ***** *.**. ***** ***** .... : 18 20 | *.*** .**** *.*** .*..* .*** : 17 21 | **..* .***. *.*** ....* .*** : 14 22 | ***** ***** .**** *.**. **** : 21 23 | ***.* ..**. *.*** ....* .*** : 14 .................................. # | 22112 11211 21222 11111 1111 | 20840 48097 06001 54539 5677