scholarly journals Combinatorics and Genus of Tropical Intersections and Ehrhart Theory

2010 ◽  
Vol 24 (1) ◽  
pp. 17-32 ◽  
Author(s):  
Reinhard Steffens ◽  
Thorsten Theobald
Keyword(s):  
Ehrhart Theory

scholarly journals The Trade-Off Between Privacy and Fidelity via Ehrhart Theory

2020 ◽  
Vol 66 (4) ◽  
pp. 2549-2569
Author(s):  
Arun Padakandla ◽  
P. R. Kumar ◽  
Wojciech Szpankowski
Keyword(s):  
Ehrhart Theory ◽  
Trade Off

scholarly journals GEOMETRIC BIJECTIONS FOR REGULAR MATROIDS, ZONOTOPES, AND EHRHART THEORY

2019 ◽  
Vol 7 ◽  
Author(s):  
SPENCER BACKMAN ◽  
MATTHEW BAKER ◽  
CHI HO YUEN
Keyword(s):  
Spanning Trees ◽  
Finite Abelian Group ◽  
Tutte Polynomial ◽  
Geometric Interpretation ◽  
Lattice Points ◽  
Combinatorial Interpretation ◽  
Ehrhart Polynomial ◽  
Ehrhart Theory ◽  
Regular Matroid ◽  
Definition Of

Let $M$ be a regular matroid. The Jacobian group $\text{Jac}(M)$ of $M$ is a finite abelian group whose cardinality is equal to the number of bases of $M$ . This group generalizes the definition of the Jacobian group (also known as the critical group or sandpile group) $\operatorname{Jac}(G)$ of a graph $G$ (in which case bases of the corresponding regular matroid are spanning trees of $G$ ). There are many explicit combinatorial bijections in the literature between the Jacobian group of a graph $\text{Jac}(G)$ and spanning trees. However, most of the known bijections use vertices of $G$ in some essential way and are inherently ‘nonmatroidal’. In this paper, we construct a family of explicit and easy-to-describe bijections between the Jacobian group of a regular matroid $M$ and bases of $M$ , many instances of which are new even in the case of graphs. We first describe our family of bijections in a purely combinatorial way in terms of orientations; more specifically, we prove that the Jacobian group of $M$ admits a canonical simply transitive action on the set ${\mathcal{G}}(M)$ of circuit–cocircuit reversal classes of $M$ , and then define a family of combinatorial bijections $\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70E}^{\ast }}$ between ${\mathcal{G}}(M)$ and bases of $M$ . (Here $\unicode[STIX]{x1D70E}$ (respectively $\unicode[STIX]{x1D70E}^{\ast }$ ) is an acyclic signature of the set of circuits (respectively cocircuits) of $M$ .) We then give a geometric interpretation of each such map $\unicode[STIX]{x1D6FD}=\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70E}^{\ast }}$ in terms of zonotopal subdivisions which is used to verify that $\unicode[STIX]{x1D6FD}$ is indeed a bijection. Finally, we give a combinatorial interpretation of lattice points in the zonotope $Z$ ; by passing to dilations we obtain a new derivation of Stanley’s formula linking the Ehrhart polynomial of $Z$ to the Tutte polynomial of $M$ .

scholarly journals Lattice Points and Simultaneous Core Partitions

10.37236/5734 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Paul Johnson
Keyword(s):  
Lattice Point ◽  
Lattice Points ◽  
Ehrhart Theory ◽  
Average Size

We apply lattice point techniques to the study of simultaneous core partitions. Our central observation is that for $a$ and $b$ relatively prime, the abacus construction identifies the set of simultaneous $(a,b)$-core partitions with lattice points in a rational simplex. We apply this result in two main ways: using Ehrhart theory, we reprove Anderson's theorem that there are $(a+b-1)!/a!b!$ simultaneous $(a,b)$-cores; and using Euler-Maclaurin theory we prove Armstrong's conjecture that the average size of an $(a,b)$-core is $(a+b+1)(a-1)(b-1)/24$. Our methods also give new derivations of analogous formulas for the number and average size of self-conjugate $(a,b)$-cores.

scholarly journals Quasisymmetric functions from combinatorial Hopf monoids and Ehrhart Theory

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Jacob White
Keyword(s):  
Commutative Algebra ◽  
Hopf Algebras ◽  
Simplicial Complexes ◽  
Hilbert Functions ◽  
Quasisymmetric Functions ◽  
Ehrhart Theory ◽  
International Audience ◽  
Hopf Monoid

International audience We investigate quasisymmetric functions coming from combinatorial Hopf monoids. We show that these invariants arise naturally in Ehrhart theory, and that some of their specializations are Hilbert functions for relative simplicial complexes. This class of complexes, called forbidden composition complexes, also forms a Hopf monoid, thus demonstrating a link between Hopf algebras, Ehrhart theory, and commutative algebra. We also study various specializations of quasisymmetric functions.

scholarly journals Ehrhart theory, modular flow reciprocity, and the Tutte polynomial

2010 ◽  
Vol 270 (1-2) ◽  
pp. 1-18 ◽  
Author(s):  
Felix Breuer ◽  
Raman Sanyal
Keyword(s):  
Tutte Polynomial ◽  
Ehrhart Theory
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