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

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!