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1. Phylogenetic trees

   https://doi.org/10.2140/jsag.2021.11.1  

   Hector Banos, Nathaniel Bushek (January 2021, The journal of software for algebra and geometry)

   null (Ed.)

   We introduce the package PhylogeneticTrees for Macaulay2, which allows users to compute phylogenetic invariants for group-based tree models. We provide some background information on phylogenetic algebraic geometry and show how the package PhylogeneticTrees can be used to calculate a generating set for a phylogenetic ideal as well as a lower bound for its dimension. Finally, we show how methods within the package can be used to compute a generating set for the join of any two ideals. 

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2. Rational growth in torus bundle groups of odd trace

   https://doi.org/10.1017/S0013091522000505  

   Choi, Seongjun; Ho, Meng-Che “Turbo”; Pengitore, Mark (November 2022, Proceedings of the Edinburgh Mathematical Society)

   Abstract A group is said to have rational growth with respect to a generating set if the growth series is a rational function. It was shown by Parry that certain torus bundle groups of even trace exhibits rational growth. We generalize this result to a class of torus bundle groups with odd trace. 

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3. Cyclic Sieving for Plane Partitions and Symmetry

   https://doi.org/10.3842/SIGMA.2020.130  

   Hopkins, Sam (December 2020, Symmetry, Integrability and Geometry: Methods and Applications)

   The cyclic sieving phenomenon of Reiner, Stanton, and White says that we can often count the fixed points of elements of a cyclic group acting on a combinatorial set by plugging roots of unity into a polynomial related to this set. One of the most impressive instances of the cyclic sieving phenomenon is a theorem of Rhoades asserting that the set of plane partitions in a rectangular box under the action of promotion exhibits cyclic sieving. In Rhoades's result the sieving polynomial is the size generating function for these plane partitions, which has a well-known product formula due to MacMahon. We extend Rhoades's result by also considering symmetries of plane partitions: specifically, complementation and transposition. The relevant polynomial here is the size generating function for symmetric plane partitions, whose product formula was conjectured by MacMahon and proved by Andrews and Macdonald. Finally, we explain how these symmetry results also apply to the rowmotion operator on plane partitions, which is closely related to promotion. 

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4. Combinatorial Results on Barcode Lattices

   https://doi.org/10.1007/s11083-024-09670-0  

   Bouquet, Alex; Vindas-Meléndez, Andrés_R (July 2024, Order)

   Abstract A barcode is a finite multiset of intervals on the real line. Jaramillo-Rodriguez (2023) previously defined a map from the space of barcodes with a fixed number of bars to a set of multipermutations, which presented new combinatorial invariants on the space of barcodes. A partial order can be defined on these multipermutations, resulting in a class of posets known as combinatorial barcode lattices. In this paper, we provide a number of equivalent definitions for the combinatorial barcode lattice, show that its Möbius function is a restriction of the Möbius function of the symmetric group under the weak Bruhat order, and show its ground set is the Jordan-Hölder set of a labeled poset. Furthermore, we obtain formulas for the number of join-irreducible elements, the rank-generating function, and the number of maximal chains of combinatorial barcode lattices. Lastly, we make connections between intervals in the combinatorial barcode lattice and certain classes of matchings. 

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5. Naturality and Mapping Class Groups in Heegaard Floer Homology

   https://doi.org/10.1090/memo/1338  

   Juhász, András; Thurston, Dylan; Zemke, Ian (September 2021, Memoirs of the American Mathematical Society)

   We show that all versions of Heegaard Floer homology, link Floer homology, and sutured Floer homology are natural. That is, they assign concrete groups to each based 3-manifold, based link, and balanced sutured manifold, respectively. Furthermore, we functorially assign isomorphisms to (based) diffeomorphisms, and show that this assignment is isotopy invariant. The proof relies on finding a simple generating set for the fundamental group of the “space of Heegaard diagrams,” and then showing that Heegaard Floer homology has no monodromy around these generators. In fact, this allows us to give sufficient conditions for an arbitrary invariant of multi-pointed Heegaard diagrams to descend to a natural invariant of 3-manifolds, links, or sutured manifolds. 

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