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A well-known conjecture of Richard Stanley posits that the $h$-vector of the independence complex of a matroid is a pure ${mathcal O}$-sequence. The conjecture has been established for various classes but is open for graphic matroids. A biconed graph is a graph with two specified `coning vertices, such that every vertex of the graph is connected to at least one coning vertex. The class of biconed graphs includes coned graphs, Ferrers graphs, and complete multipartite graphs. We study the $h$-vectors of graphic matroids arising from biconed graphs, providing a combinatorial interpretation of their entries in terms of `edge-rooted forests of the underlying graph. This generalizes constructions of Kook and Lee who studied the Mobius coinvariant (the last nonzero entry of the $h$-vector) of graphic matroids of complete bipartite graphs. We show that allowing for partially edge-rooted forests gives rise to a pure multicomplex whose face count recovers the $h$-vector, establishing Stanleys conjecture for this class of matroids.
We show that families of action graphs, with initial graphs which are linear of varying length, give rise to self-convolutions of the Catalan sequence. We prove this result via a comparison with planar rooted forests with a fixed number of trees.
Given an infinite field $mathbb{k}$ and a simplicial complex $Delta$, a common theme in studying the $f$- and $h$-vectors of $Delta$ has been the consideration of the Hilbert series of the Stanley--Reisner ring $mathbb{k}[Delta]$ modulo a generic lin
In this article, we discuss the vertex decomposability of three well-studied simplicial complexes associated to forests. In particular, we show that the bounded degree complex of a forest and the complex of directed trees of a multidiforest are verte
Building off recent work of Garg and Peng, we continue the investigation into classical and consecutive pattern avoidance in rooted forests, resolving some of their conjectures and questions and proving generalizations whenever possible. Through exte
We prove that for $k in mathbb{N}$ and $d leq 2k+2$, if a graph has maximum average degree at most $2k + frac{2d}{d+k+1}$, then $G$ decomposes into $k+1$ pseudoforests, where one of the pseudoforests has all connected components having at most $d$ edges.