2020/06/29 by Michael Fuchs, Fuchs, Michael, Bernhard Gittenberger +3 · 7 citations
Biochemistry, Genetics and Molecular Biology · Mathematics · #05A15 #92B05 #Combinatorics #Combinatorics (math.CO) #Counting problem #Discrete mathematics #Enumeration #Exponential function #Exponential growth #FOS: Mathematics #Fractal and DNA sequence analysis #Generating function #Genetic diversity and population structure #Genomics and Phylogenetic Studies #Mathematical analysis #Mathematics #Tree (set theory) #Zhàng #math.CO #msc:05A15 #msc:92B05
paper · pdf · doi:10.48550/arxiv.2006.15784
published in arXiv (Cornell University) (Cornell University) · revised version; 20 pages
openalex publication_date 2020/06/29 · openalex created_date 2020/07/02 · arxiv created 2021/03/04 · arxiv updated 2021/03/05 · openalex updated_date 2026/07/28
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.