For every finite matroid M of rank r, let Delta(M) be its independence complex and let h(M)=(h_0,…,h_r) be defined by ∑_i=0^rf_i-1(Delta(M))(t-1)^r-i=∑_i=0^rh_i t^r-i, where f_-1(Delta(M))=1. Then h(M) is a pure O-sequence: there exists a finite pure multicomplex Gamma of monomials such that h_i=|u∈Gamma:deg u=i| for every 0≤ i≤ r.

Stanley’s matroid h-vector conjecture

Stanley’s conjecture asserts that every matroid $h$ -vector is a pure $O$ -sequence. The general problem was unresolved in the 2013 survey literature and remains unresolved after the latest advance.

Known results

Matroid complexes of Krull dimension at most $2$ , including all $h$ -vectors of the form $(1,h_1,h_2,h_3)$ , were handled by H. T. Hà, E. Stokes, and F. Zanello (2013).

September 2026 fibre-system criterion

SuHo Oh’s paper introduces a sufficient fibre-system criterion for constructing the required pure multicomplexes and verifies it for coned, biconed, and triconed graphs, several named graphs, matroids of rank at most $3$ , corank $2$ , and uniform matroids. It also exhibits a twelve-vertex graph without a fibre system; this is substantial claimed progress, not a proof for general matroids.

Current status (as of September 2026): The conjecture is established in the cited low-dimensional cases and several additional families, while the general matroid case remains open; Oh’s new criterion is reported as progress but is not independently verified here.

Stanley’s matroid h-vector conjecture

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Additional references

Progress summary

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