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1. Ehrhart theory of paving and panhandle matroids

   https://doi.org/10.1515/advgeom-2023-0020  

   Hanely, Derek; Martin, Jeremy L.; McGinnis, Daniel; Miyata, Dane; Nasr, George D.; Vindas-Meléndez, Andrés R.; Yin, Mei (October 2023, Advances in Geometry)

   Abstract We show that the base polytopeP_Mof any paving matroidMcan be systematically obtained from a hypersimplex by slicing off certain subpolytopes, namely base polytopes of lattice path matroids corresponding to panhandle-shaped Ferrers diagrams. We calculate the Ehrhart polynomials of these matroids and consequently write down the Ehrhart polynomial ofP_M, starting with Katzman’s formula for the Ehrhart polynomial of a hypersimplex. The method builds on and generalizes Ferroni’s work on sparse paving matroids. Combinatorially, our construction corresponds to constructing a uniform matroid from a paving matroid by iterating the operation ofstressed-hyperplane relaxationintroduced by Ferroni, Nasr and Vecchi, which generalizes the standard matroid-theoretic notion of circuit-hyperplane relaxation. We present evidence that panhandle matroids are Ehrhart positive and describe a conjectured combinatorial formula involving chain forests and Eulerian numbers from which Ehrhart positivity of panhandle matroids will follow. As an application of the main result, we calculate the Ehrhart polynomials of matroids associated with Steiner systems and finite projective planes, and show that they depend only on their design-theoretic parameters: for example, while projective planes of the same order need not have isomorphic matroids, their base polytopes must be Ehrhart equivalent. 

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2. Stellahedral geometry of matroids

   https://doi.org/10.1017/fmp.2023.24  

   Eur, Christopher; Huh, June; Larson, Matt (January 2023, Forum of Mathematics, Pi)

   Abstract We use the geometry of the stellahedral toric variety to study matroids. We identify the valuative group of matroids with the cohomology ring of the stellahedral toric variety and show that valuative, homological and numerical equivalence relations for matroids coincide. We establish a new log-concavity result for the Tutte polynomial of a matroid, answering a question of Wagner and Shapiro–Smirnov–Vaintrob on Postnikov–Shapiro algebras, and calculate the Chern–Schwartz–MacPherson classes of matroid Schubert cells. The central construction is the ‘augmented tautological classes of matroids’, modeled after certain toric vector bundles on the stellahedral toric variety. 

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3. A Two-Stage Constrained Submodular Maximization

   https://doi.org/10.1007/978-3-030-27195-4_30  

   Ruiqi Yang, Shuyang Gu (August 2019, AAIM 2019)

   We consider a two-stage submodular maximization under p-matroid (or p-extendible) constraints. In the model, we are given a collection of submodular functions and some p-matroid (or extendible) system constraints for each of these functions, one need to choose a representative set with a cardinality constraint and simultaneously select a series of subsets that are restricted to the representative set for all functions, the aim is to maximize the average of the summarization of these function values. We extend the two-stage submodular maximization under single matroid to handle p-matroid (or p-extendible) constraints, and derive constant approximation ratio algorithms for the two problems, respectively. In the end, we empirically demonstrate the efficiency of our method on some datasets. 

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4. $h$ -vector inequalities under weak maps

   https://doi.org/10.5802/alco.409  

   Liu, Gaku; Mason, Alexander (April 2025, Algebraic Combinatorics)

   Munemasa, Akihiro; Murai, Satoshi; Rhoades, Brendon; Speyer, David; Maldeghem, Hendrik Van (Ed.)

   We study the behavior of $h$-vectors associated to matroid complexes under weak maps, or inclusions of matroid polytopes. Specifically, we show that the $h$-vector of the order complex of the lattice of flats of a matroid is component-wise non-increasing under a weak map. This result extends to the flag $h$-vector. We note that the analogous result also holds for independence complexes and rank-preserving weak maps. 

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5. Equivariant Cohomology and Conditional Oriented Matroids

   https://doi.org/10.1093/imrn/rnad025  

   Dorpalen-Barry, Galen; Proudfoot, Nicholas; Wang, Jidong (May 2024, International Mathematics Research Notices)

   Abstract We give a cohomological interpretation of the Heaviside filtration on the Varchenko–Gelfand ring of a pair $$({\mathcal{A}},{\mathcal{K}})$$, where $${\mathcal{A}}$$ is a real hyperplane arrangement and $${\mathcal{K}}$$ is a convex open subset of the ambient vector space. This builds on work of the first author, who studied the filtration from a purely algebraic perspective, as well as work of Moseley, who gave a cohomological interpretation in the special case where $${\mathcal{K}}$$ is the ambient vector space. We also define the Gelfand–Rybnikov ring of a conditional oriented matroid, which simultaneously generalizes the Gelfand–Rybnikov ring of an oriented matroid and the aforementioned Varchenko–Gelfand ring of a pair. We give purely combinatorial presentations of the ring, its associated graded, and its Rees algebra. 

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