# The h-vector of a matroid complex, paving matroids and the chip firing game

A non-empty set of monomials $\Sigma$ is a multicomplex if for any monomial $z$ in $\Sigma$ and a monomial $z'$ such that $z'|z$, we have that $z'$ also belongs to $\Sigma$. A multicomplex $\Sigma$ is called pure if all its maximal elements have the same degree. This notion is a generalization of the simplicial complex, and several invariants extend directly, as the $f$-vector of a multicomplex, which is the vector that lists the monomials grouped by degrees. A non-empty set of monomials $\Sigma$ is a multicomplex if for any monomial $z$ in $\Sigma$ and a monomial $z'$ such that $z'|z$, we have that $z'$ also belongs to $\Sigma$. A multicomplex $\Sigma$ is called pure if all its maximal elements have the same degree. This notion is a generalization of the simplicial complex, and several invariants extend directly, as the $f$-vector of a multicomplex, which is the vector that lists the monomials grouped by degrees. The relevance of multicomplexes in matroid theory is partly due to a 1977 Richard Stanley conjecture that says that the $h$-vector of a matroid complex is the $f$-vector of a pure multicomplex. This has been proved for several families of matroids. In this talk, we review some results of Stanley’s conjecture, mainly for paving and cographic matroids. A paving matroid is one in which all its circuits have a size of at least the rank of the matroid. While, the chip firing game is a solitaire game played on a connected graph $G$ that surprisingly is related to the $h$-vector of the bond matroid of $G$.