Ehrhart theory of paving and panhandle matroids
-
Derek Hanely
,
Andrés R. Vindas-Meléndez
Abstract
We show that the base polytope PM of any paving matroid M can 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 of PM, 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 of stressed-hyperplane relaxation introduced 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.
Funding statement: Martin is partially supported by Simons Collaboration Grant #315347. McGinnis is partially supported by the National Science Foundation under grant DMS-1839918 (RTG). Miyata is partially supported by the National Science Foundation under grants DMS-1954050, DMS-2039316 (RTG), and DMS-2053243 (FRG). Nasr was partially supported by the National Science Foundation under grant DMS-2053243 (FRG). Vindas-Meléndez is partially supported by the National Science Foundation under Award DMS-2102921. Yin is partially supported by the University of Denver’s Faculty Research Fund 84688-145601.
Acknowledgement
The authors thank Mohsen Aliabadi, Margaret Bayer, Matthias Beck, Federico Castillo, Luis Ferroni, Joseph Kung, and James Oxley for fruitful correspondence. This work was initiated at the 2021 Graduate Research Workshop in Combinatorics.
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© 2023 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Exploring tropical differential equations
- On the bond polytope
- Universal convex covering problems under translations and discrete rotations
- Ehrhart theory of paving and panhandle matroids
- A partial compactification of the Bridgeland stability manifold
- The geometry of discrete L-algebras
- Projective self-dual polygons in higher dimensions
Articles in the same Issue
- Frontmatter
- Exploring tropical differential equations
- On the bond polytope
- Universal convex covering problems under translations and discrete rotations
- Ehrhart theory of paving and panhandle matroids
- A partial compactification of the Bridgeland stability manifold
- The geometry of discrete L-algebras
- Projective self-dual polygons in higher dimensions
