h-Vectors of Small Matroid Complexes

Programs, Tables, and Statistics

Introduction

In the tables following, we present the h-vectors for matroids on at most nine elements, as well as monomial families associated with the pure O-sequences corresponding to each of these matroid h-vectors. We have also included the computer programs used to perform the necessary computations and the number of h-vectors of each rank (0 through 9) and co-rank (0 through 9).

The Programs

h-Vector Computer
Based on David Haws's original code, this program takes as input matroids from Gordon Royle and Dillon Mayhew's list of matroids on at most nine elements, and outputs their h-vectors.
O-Sequence Generator
This perl code was used to generate large numbers of O-sequences for a particular rank and co-rank. The following are the perl codes used to generate the for loops necessary for each step:
- For Loop Generator
  This perl code was used to generate the for loops which build monomial families in the O-sequence generator.
- LaTeX Loop Generator
  This perl code was used to generator the loops which printed the relevant information to file in the LaTeX format.
boxy
This Maple code uses Barvinok generating functions to generate O-sequences given a particular set of top-level vertices.

Tables and Statistics

Please click on the box corresponding to the appropriate rank and co-rank of the matroid to view the h-vectors and accompanying top-level monomials. The numbers in each of the boxes correspond to the number of distinct matroid h-vectors of this type. Note that the cases of ranks 1, 2, and 3, and co-ranks 1 and 2 have already been verified, so we do not provide monomials for these cases. Further note that we have not listed monomials for matroids with co-loops: a matroid having j co-loops has an h-vector with j zeroes at the end, and the non-zero entries correspond to the h-vector of the same matroid with all co-loops contracted. Since this new matroid also has a ground set of at most nine elements, a family of monomials will be provided elsewhere in the table.

Finally, if the rank plus co-rank is greater than or equal to 10, we have no information on the quantities of matroid h-vectors, and have indicated this with '--'.

Number of Distinct Matroid h-Vectors

Total Number of Matroids

rank\co-rank	0	1	2	3	4	5	6	7	8	9
0	0	1	1	1	1	1	1	1	1	1
1	1	1	1	1	1	1	1	1	1	--
2	1	2	4	6	8	12	17	20	--	--
3	1	3	9	22	49	101	196	--	--	--
4	1	4	18	67	244	816	--	--	--	--
5	1	5	31	186	1132	--	--	--	--	--
6	1	6	51	489	--	--	--	--	--	--
7	1	7	79	--	--	--	--	--	--	--
8	1	8	--	--	--	--	--	--	--	--
9	1	--	--	--	--	--	--	--	--	--

rank\co-rank	0	1	2	3	4	5	6	7	8	9
0	0	1	1	1	1	1	1	1	1	1
1	9	8	7	6	5	4	3	2	1	--
2	8	14	24	30	40	42	42	29	--	--
3	7	18	45	100	210	434	950	--	--	--
4	6	20	72	255	1664	189274	--	--	--	--
5	5	20	93	576	189889	--	--	--	--	--
6	4	18	102	1217	--	--	--	--	--	--
7	3	14	79	--	--	--	--	--	--	--
8	2	8	--	--	--	--	--	--	--	--
9	1	--	--	--	--	--	--	--	--	--

Comments, questions, or concerns about the programs or tables can be sent to Yvonne Kemper (ykemper AT math DOT ucdavis DOT edu). Thanks!