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Register a new userCounting Phylogenetic Networks with Few Reticulation Vertices: Exact Enumeration and Corrections
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In previous work, we gave asymptotic counting results for the number of tree-child and normal networks with $k$ reticulation vertices and explicit exponential generating functions of the counting sequences for $k=1,2,3$. The purpose of this note is two-fold. First, we make some corrections to our previous approach which overcounted the above numbers and thus gives erroneous exponential generating functions (however, the overcounting does not effect our asymptotic counting results). Secondly, we use our (corrected) exponential generating functions to derive explicit formulas for the number of tree-child and normal networks with $k=1,2,3$ reticulation vertices. This re-derives recent results of Carona and Zhang, answers their question for normal networks with $k=2$, and adds new formulas in the case $k=3$.
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Tree-child networks, one of the prominent network classes in phylogenetics, have been introduced for the purpose of modeling reticulate evolution. Recently, the first author together with Gittenberger and Mansouri (2019) showed that the number ${rm TC}_{ell,k}$ of tree-child networks with $ell$ leaves and $k$ reticulation vertices has the first-order asymptotics [ {rm TC}_{ell,k}sim c_kleft(frac{2}{e}right)^{ell}ell^{ell+2k-1},qquad (ellrightarrowinfty). ] Moreover, they also computed $c_k$ for $k=1,2,$ and $3$. In this short note, we give a second approach to the above result which is based on a recent (algorithmic) approach for the counting of tree-child networks due to Cardona and Zhang (2020). This second approach is also capable of giving a simple, closed-form expression for $c_k$ for all $kgeq 0$.
For $ngeq 3$, let $r=r(n)geq 3$ be an integer. A hypergraph is $r$-uniform if each edge is a set of $r$ vertices, and is said to be linear if two edges intersect in at most one vertex. In this paper, the number of linear $r$-uniform hypergraphs on $ntoinfty$ vertices is determined asymptotically when the number of edges is $m(n)=o(r^{-3}n^{ frac32})$. As one application, we find the probability of linearity for the independent-edge model of random $r$-uniform hypergraph when the expected number of edges is $o(r^{-3}n^{ frac32})$. We also find the probability that a random $r$-uniform linear hypergraph with a given number of edges contains a given subhypergraph.
Reticulate evolutionary processes result in phylogenetic histories that cannot be modeled using a tree topology. Here, we apply methods from topological data analysis to molecular sequence data with reticulations. Using a simple example, we demonstrate the correspondence between nontrivial higher homology and reticulate evolution. We discuss the sensitivity of the standard filtration and show cases where reticulate evolution can fail to be detected. We introduce an extension of the standard framework and define the median complex as a construction to recover signal of the frequency and scale of reticulate evolution by inferring and imputing putative ancestral states. Finally, we apply our methods to two datasets from phylogenetics. Our work expands on earlier ideas of using topology to extract important evolutionary features from genomic data.
The main contribution of this paper is a new column-by-column method for the decomposition of generating functions of convex polyominoes suitable for enumeration with respect to various statistics including but not limited to interior vertices, boundary vertices of certain degrees, and outer site perimeter. Using this decomposition, among other things, we show that A) the average number of interior vertices over all convex polyominoes of perimeter $2n$ is asymptotic to $frac{n^2}{12}+frac{nsqrt{n}}{3sqrt{pi}} -frac{(21pi-16)n}{12pi}.$ B) the average number of boundary vertices with degree two over all convex polyominoes of perimeter $2n$ is asymptotic to $frac{n+6}{2}+frac{1}{sqrt{pi n}}+frac{(16-7pi)}{4pi n}.$ Additionally, we obtain an explicit generating function counting the number of convex polyominoes with $n$ boundary vertices of degrees at most three and show that this number is asymptotic to $ frac{n+1}{40}left(frac{3+sqrt{5}}{2}right)^{n-3} +frac{sqrt[4]{5}(2-sqrt{5})}{80sqrt{pi n}}left(frac{3+sqrt{5}}{2}right)^{n-2}. $ Moreover, we show that the expected number of the boundary vertices of degree four over all convex polyominoes with $n$ vertices of degrees at most three is asymptotically $ frac{n}{sqrt{5}}-frac{sqrt[4]{125}(sqrt{5}-1)sqrt{n}}{10sqrt{pi}}. $ C) the number of convex polyominoes with the outer-site perimeter $n$ is asymptotic to $frac{3(sqrt{5}-1)}{20sqrt{pi n}sqrt[4]{5}}left(frac{3+sqrt{5}}{2}right)^n,$ and show the expected number of the outer-site perimeter over all convex polyominoes with perimeter $2n$ is asymptotic to $frac{25n}{16}+frac{sqrt{n}}{4sqrt{pi}}+frac{1}{8}.$ Lastly, we prove that the expected perimeter over all convex polyominoes with the outer-site perimeter $n$ is asymptotic to $sqrt[4]{5}n$.
In this paper we enumerate the cardinalities for the set of all vertices of outdegree $ge k$ at level $ge ell$ among all rooted ordered $d$-trees with $n$ edges. Our results unite and generalize several previous works in the literature.
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