Note: It was previously known that these matroids have h-vectors which are pure O-sequences, thus we have not applied the computer algorithms to generate monomials for them. See our paper for details.

h-vector
(1,2,0,0,0,0,0)
(1,2,1,0,0,0,0)
(1,2,2,0,0,0,0)
(1,2,2,1,0,0,0)
(1,2,2,2,0,0,0)
(1,2,2,2,1,0,0)
(1,2,2,2,2,0,0)
(1,2,2,2,2,1,0)
(1,2,2,2,2,2,0)
(1,2,2,2,2,2,1)
(1,2,2,2,2,2,2)
(1,2,3,0,0,0,0)
(1,2,3,2,0,0,0)
(1,2,3,2,1,0,0)
(1,2,3,3,0,0,0)
(1,2,3,3,2,0,0)
(1,2,3,3,2,1,0)
(1,2,3,3,3,0,0)
(1,2,3,3,3,2,0)
(1,2,3,3,3,2,1)
(1,2,3,3,3,3,0)
(1,2,3,3,3,3,2)
(1,2,3,3,3,3,3)
(1,2,3,4,0,0,0)
(1,2,3,4,2,0,0)
(1,2,3,4,3,0,0)
(1,2,3,4,3,2,0)
(1,2,3,4,3,2,1)
(1,2,3,4,4,0,0)
(1,2,3,4,4,2,0)
(1,2,3,4,4,3,0)
(1,2,3,4,4,3,2)
(1,2,3,4,4,4,0)
(1,2,3,4,4,4,2)
(1,2,3,4,4,4,3)
(1,2,3,4,4,4,4)
(1,2,3,4,5,0,0)
(1,2,3,4,5,3,0)
(1,2,3,4,5,4,0)
(1,2,3,4,5,4,2)
(1,2,3,4,5,4,3)
(1,2,3,4,5,5,0)
(1,2,3,4,5,5,3)
(1,2,3,4,5,5,4)
(1,2,3,4,5,5,5)
(1,2,3,4,5,6,0)
(1,2,3,4,5,6,3)
(1,2,3,4,5,6,4)
(1,2,3,4,5,6,5)
(1,2,3,4,5,6,6)
(1,2,3,4,5,6,7)