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.
In this paper, we express the generating function of a combination of cranks of partitions that belong to congruence classes 3, 4, 5, 6, and 7 modulo 15, in terms of the Eisenstein series. Our results rely on the observation that the generating function of the combination we consider is a weight 1 modular form on Gamma 0(225) with the non-trivial modulo 3 primitive Dirichlet character. We obtain the modularity result by analyzing the transformation properties of related Klein forms. The main results are proven using the fact that the corresponding space is generated by Eisenstein series. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
Let M be a simple connected binary matroid, and let e and f be distinct elements of M. It is well known that, when the only circuits containing e are spanning, M is a circuit with at least three elements. This paper proves that if every circuit containing {e,f} is spanning, then the canonical tree decomposition of M is a path in which each vertex is labeled by a circuit, a copy of U_1,3, or a binary spike having one non-tip element deleted.
This paper establishes connections between three of the most prominent metrics used in the analysis of persistence diagrams in topological data analysis: the bottleneck distance, Patel's erosion distance, and Bubenik's landscape distance. Our main result shows that the erosion and landscape distances are equal, thereby bridging the former's natural category-theoretic interpretation with the latter's computationally convenient structure. The proof utilizes the category with a flow framework of de Silva et al., and leads to additional insights into the structure of persistence landscapes. Our equivalence result is applied to prove several results on the geometry of the erosion distance. We show that the erosion distance is not a length metric, and that its intrinsic metric is the bottleneck distance. We also show that the erosion distance does not coarsely embed into any Hilbert space, even when restricted to persistence diagrams arising from degree-0 persistent homology. Moreover, we show that erosion distance agrees with bottleneck distance on this subspace, so that our non-embeddability theorem generalizes several results in the recent literature.
