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1. Model structures on finite total orders

   https://doi.org/10.1007/s00209-023-03287-6  

   Balchin, Scott; Ormsby, Kyle; Osorno, Angélica M.; Roitzheim, Constanze (July 2023, Mathematische Zeitschrift)

   Abstract We initiate the study of model structures on (categories induced by) lattice posets, a subject we dub homotopical combinatorics . In the case of a finite total order [ n ], we enumerate all model structures, exhibiting a rich combinatorial structure encoded by Shapiro’s Catalan triangle. This is an application of previous work of the authors on the theory of $$N_\infty $$ N ∞ -operads for cyclic groups of prime power order, along with new structural insights concerning extending choices of certain model structures on subcategories of [ n ]. 

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2. Realizing the s-Permutahedron via Flow Polytopes

   González_D'León, Rafael_S; Morales, Alejandro_H; Philippe, Eva; Tamayo_Jiménez, Daniel; Yip, Martha (July 2024, Séminaire Lotharingien de Combinatoire)

   In 2019, Ceballos and Pons introduced the s-weak order on s-decreasing trees, for any weak composition s. They proved its lattice structure and conjectured that it could be realized as the 1-skeleton of a polyhedral subdivision of a zonotope of dimension n-1. We answer their conjecture in the case where s is a (strict) composition by providing three geometric realizations of the s-permutahedron. The first one is the dual graph of a triangulation of a flow polytope of high dimension. The second, obtained using the Cayley trick, is the dual graph of a fine mixed subdivision of a sum of hypercubes that has the conjectured dimension. The third, obtained using tropical geometry, is the 1-skeleton of a polyhedral complex for which we can provide explicit coordinates of the vertices and whose support is a permutahedron as conjectured. 

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3. 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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4. Mechanical properties of additively manufactured variable lattice structures of Ti6Al4V

   https://doi.org/doi.org/10.1016/j.msea.2021.140925  

   Traxel, Kellen; Groden, Cory; Valladares, Jesus; Bandyopadhyay, Amit (March 2021, Materials science engineering)

   null (Ed.)

   Engineered micro- and macro-structures via additive manufacturing (AM) or 3D-Printing can create structurally varying properties in a part, which is difficult via traditional manufacturing methods. Herein we have utilized powder bed fusion-based selective laser melting (SLM) to fabricate variable lattice structures of Ti6Al4V with uniquely designed unit cell configurations to alter the mechanical performance. Five different configurations were designed based on two natural crystal structures – hexagonal closed packed (HCP) and body centered cubic (BCC). Under compressive loading, as much as 74% difference was observed in compressive strength, and 71% variation in elastic modulus, with all samples having porosities in a similar range of 53 to 65%, indicating the influence of macro-lattice designs alone on mechanical properties. Failure analysis of the fracture surfaces helped with the overall understanding of how configurational effects and unit cell design influences mechanical properties of these samples. Our work highlights the ability to leverage advanced manufacturing techniques to tailor the structural performance of multifunctional components. 

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5. Assouad–Nagata dimension of minor‐closed metrics

   https://doi.org/10.1112/plms.70032  

   Liu, Chun‐Hung (March 2025, Proceedings of the London Mathematical Society)

   Abstract Assouad–Nagata dimension addresses both large‐ and small‐scale behaviors of metric spaces and is a refinement of Gromov's asymptotic dimension. A metric space is a minor‐closed metric if there exists an (edge‐)weighted graph satisfying a fixed minor‐closed property such that the underlying space of is the vertex‐set of , and the metric of is the distance function in . Minor‐closed metrics naturally arise when removing redundant edges of the underlying graphs by using edge‐deletion and edge‐contraction. In this paper, we determine the Assouad–Nagata dimension of every minor‐closed metric. Our main theorem simultaneously generalizes known results about the asymptotic dimension of ‐minor free unweighted graphs and about the Assouad–Nagata dimension of complete Riemannian surfaces with finite Euler genus (Bonamy et al., Asymptotic dimension of minor‐closed families and Assouad–Nagata dimension of surfaces,JEMS(2024)). 

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