The study of simultaneous core partitions has attracted considerable attention over the past two decades. In this paper, we determine the largest size of (s,s+3)-core partitions with distinct parts for s≡1 or 2(mod3), and we show that the number of such partitions attaining the largest size is always 1. In particular, our result for s≡1(mod3) confirms a conjecture of Straub.
Phylogenetic networks provide a powerful framework for modeling reticulate evolutionary processes such as hybridization, recombination, and horizontal gene transfer. However, inferring phylogenetic networks remains challenging because the vast space of possible networks is difficult to characterize. Here, we prove that, when k=o(n), the number of binary phylogenetic networks with k reticulations on n taxa is asymptotically(nk)2n+k−1/2nn+k−1e−n. We further show that the class of normal networks has the same asymptotic count in this range of k. These results resolve open problems on counting binary phylogenetic networks.
