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Register a new userThe Combinatorics of Directed Planar Trees
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We give a geometric realization of the polyhedra governed by the structure of associative algebras with co-inner products, or more precisely, governed by directed planar trees. Our explicit realization of these polyhedra, which include the associahedra in a special case, shows in particular that these polyhedra are homeomorphic to balls. We also calculate the number of vertices of the lowest generalized associahedra, giving appropriate generalizations of the Catalan numbers.
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In 2001, Komlos, Sarkozy and Szemeredi proved that, for each $alpha>0$, there is some $c>0$ and $n_0$ such that, if $ngeq n_0$, then every $n$-vertex graph with minimum degree at least $(1/2+alpha)n$ contains a copy of every $n$-vertex tree with maximum degree at most $cn/log n$. We prove the corresponding result for directed graphs. That is, for each $alpha>0$, there is some $c>0$ and $n_0$ such that, if $ngeq n_0$, then every $n$-vertex directed graph with minimum semi-degree at least $(1/2+alpha)n$ contains a copy of every $n$-vertex oriented tree whose underlying maximum degree is at most $cn/log n$. As with Komlos, Sarkozy and Szemeredis theorem, this is tight up to the value of $c$. Our result improves a recent result of Mycroft and Naia, which requires the oriented trees to have underlying maximum degree at most $Delta$, for any constant $Delta$ and sufficiently large $n$. In contrast to the previous work on spanning trees in dense directed or undirected graphs, our methods do not use Szemeredis regularity lemma.
These notes are a written version of my talk given at the CARMA workshop in June 2017, with some additional material. I presented a few concepts that have recently been used in the computation of tree-level scattering amplitudes (mostly using pure spinor methods but not restricted to it) in a context that could be of interest to the combinatorics community. In particular, I focused on the appearance of {it planar binary trees} in scattering amplitudes and presented some curious identities obeyed by related objects, some of which are known to be true only via explicit examples.
Let $G$ be a finitely generated group acting faithfully and properly discontinuously by homeomorphisms on a planar surface $X subseteq mathbb{S}^2$. We prove that $G$ admits such an action that is in addition co-compact, provided we can replace $X$ by another surface $Y subseteq mathbb{S}^2$. We also prove that if a group $H$ has a finitely generated Cayley (multi-)graph $C$ covariantly embeddable in $mathbb{S}^2$, then $C$ can be chosen so as to have no infinite path on the boundary of a face. The proofs of these facts are intertwined, and the classes of groups they define coincide. In the orientation-preserving case they are exactly the (isomorphism types of) finitely generated Kleinian function groups. We construct a finitely generated planar Cayley graph whose group is not in this class. In passing, we observe that the Freudenthal compactification of every planar surface is homeomorphic to the sphere.
Background: We study the sparsification of dynamic programming folding algorithms of RNA structures. Sparsification applies to the mfe-folding of RNA structures and can lead to a significant reduction of time complexity. Results: We analyze the sparsification of a particular decomposition rule, $Lambda^*$, that splits an interval for RNA secondary and pseudoknot structures of fixed topological genus. Essential for quantifying the sparsification is the size of its so called candidate set. We present a combinatorial framework which allows by means of probabilities of irreducible substructures to obtain the expected size of the set of $Lambda^*$-candidates. We compute these expectations for arc-based energy models via energy-filtered generating functions (GF) for RNA secondary structures as well as RNA pseudoknot structures. For RNA secondary structures we also consider a simplified loop-energy model. This combinatorial analysis is then compared to the expected number of $Lambda^*$-candidates obtained from folding mfe-structures. In case of the mfe-folding of RNA secondary structures with a simplified loop energy model our results imply that sparsification provides a reduction of time complexity by a constant factor of 91% (theory) versus a 96% reduction (experiment). For the full loop-energy model there is a reduction of 98% (experiment).
We give a brief overview of the life and combinatorics of Jeff Remmel, a mathematician with successful careers in both logic and combinatorics.
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