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
(1,1,0,0,0,0,0)
(1,1,1,0,0,0,0)
(1,1,1,1,0,0,0)
(1,1,1,1,1,0,0)
(1,1,1,1,1,1,0)
(1,1,1,1,1,1,1)