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Register a new userBiconed graphs, edge-rooted forests, and h-vectors of matroid complexes
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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.
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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 linear system of parameters $Theta$. Historically, these computations have been restricted to special classes of complexes (most typically triangulations of spheres or manifolds). We provide a compact topological expression of $h_{d-1}^mathfrak{a}(Delta)$, the dimension over $mathbb{k}$ in degree $d-1$ of $mathbb{k}[Delta]/(Theta)$, for any complex $Delta$ of dimension $d-1$. In the process, we provide tools and techniques for the possible extension to other coefficients in the Hilbert series.
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 vertex decomposable. We then prove that the non-cover complex of a forest is either contractible or homotopy equivalent to a sphere. Finally, we provide a complete characterization of forests whose non-cover complexes are vertex decomposable.
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 extensions of the forest Simion-Schmidt bijection introduced by Anders and Archer, we demonstrate a new family of forest-Wilf equivalences, completing the classification of forest-Wilf equivalence classes for sets consisting of a pattern of length 3 and a pattern of length at most $5$. We also find a new family of nontrivial c-forest-Wilf equivalences between single patterns using the forest analogue of the Goulden-Jackson cluster method, showing that a $(1-o(1))^n$-fraction of patterns of length $n$ satisfy a nontrivial c-forest-Wilf equivalence and that there are c-forest-Wilf equivalence classes of patterns of length $n$ of exponential size. Additionally, we consider a forest analogue of super-strong-c-Wilf equivalence, introduced for permutations by Dwyer and Elizalde, showing that super-strong-c-forest-Wilf equivalences are trivial by enumerating linear extensions of forest cluster posets. Finally, we prove a forest analogue of the Stanley-Wilf conjecture for avoiding a single pattern as well as certain other sets of patterns. Our techniques are analytic, easily generalizing to different types of pattern avoidance and allowing for computations of convergent lower bounds of the forest Stanley-Wilf limit in the cases covered by our result. We end with several open questions and directions for future research, including some on the limit distributions of certain statistics of pattern-avoiding forests.
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.
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