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)
(1,2,1,0)
(1,2,2,0)
(1,2,2,1)
(1,2,2,2)
(1,2,3,0)
(1,2,3,2)
(1,2,3,3)
(1,2,3,4)