Introduction

Data files

• Isomorphism classes of matroids with prescribed rank and size from the database of matroids. Each matroid is encoded as a 0/1 list whose entries correspond to the $d$-element subsets of $[n]$ under revlex order.
  • File matroidsnXrY.sage contains matroids of rank $Y$ on a ground set of size $X$.
  • Matroid data files
• Lists of unique sets of restricted $h$-vectors, together with an ordered matroid realizing each set.
  • File hvecsnXrY.sage contains representatives of sets of restricted $h$-vectors and corresponding matroids in rank $Y$ with a ground set of size $X$.
  • h-vector data files

Sage files

• Here are the remaining files to manipulate matroids and construct their corresponding pure multicomplexes.
• constructMatroids.sage: Code to convert 0/1 lists of $d$-element sets from the files matroidsnXrY.sage into lists of bases. For each isomorphism class of matroids, code constructs all possible based matroids, computes the set of restricted $h$-vectors, and saves a list of distinct sets of restricted $h$-vectors that need to be checked. The resulting lists are output to the files hvecsnXrY.sage. In those files, the lists candidateHnXrY store the lists of candidate restricted $h$-vectors, and the lists candidateComplexesnXrY store the corresponding lists of matroids that realize those restricted $h$-vectors.
• verifyLemmas.sage: Code to verify the lemmas regarding the combinatorial structure of the lists of restricted $h$-vectors. These are Lemmas 4.1, 5.1, 5.2 in our paper.
• constructMonomialsRank3.sage and constructMonomialsRank4.sage: Algorithms that construct the families of monomials corresponding to each distinct set of restricted $h$-vectors. This is the implementation of Algorithms 4.2 and 5.3 from our paper. Monomial families are stored as lists of exponent vectors.
• verifyPureOSequence.sage: Code to verify that each monomial family output by the algorithms in constructMonomialsRankX.sage is a pure multicomplex.
• verifyAlgorithm.sage:
  • Inputs all families of restricted $h$-vectors that need to be checked into the algorithms in the files constructMonomialsRankX.sage.
  • Uses verifyPureOSequence.sage to check that the output of the algorithms are pure multicomplexes.
  • Verifies that the $h$-vector of the matroid that is input agrees with the $F$-vector of the multicomplex that is output.
• matroidOperations.sage: Back end code used to manipulate based matroids in the initialization of Algorithm 5.3.

Other links

• Database of matroids of Matsumoto, Moriyama, Imai, and Bremner.
• A link to our paper on the arXiv.
• Sage cloud
• Sage home