Pure $\mathcal{O}$-Sequences arising from $2$-Dimensional PS Ear-Decomposable Simplicial Complexes

Abstract

We show that the $h$-vector of a $2$-dimensional PS ear-decomposable simplicial complex is a pure $\mathcal{O}$-sequence. This provides a strengthening of Stanley's conjecture for matroid $h$-vectors in rank $3$.  Our approach modifies the approach of combinatorial shifting for arbitrary simplicial complexes to the setting of $2$-dimensional PS ear-decomposable complexes, which allows us to greedily construct pure multicomplex corresponding to each PS ear-decomposable complex.