The equivariant Ehrhart theory of the permutahedron

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 $M$ .

Download Full-text

Additive Number Theory and Inequalities in Ehrhart Theory

International Mathematics Research Notices ◽

10.1093/imrn/rnv186 ◽

2015 ◽

Vol 2016 (5) ◽

pp. 1497-1540 ◽

Cited By ~ 5

Author(s):

Alan Stapledon

Keyword(s):

Number Theory ◽

Additive Number Theory ◽

Ehrhart Theory ◽

Additive Number

Download Full-text

An Invitation to Ehrhart Theory: Polyhedral Geometry and its Applications in Enumerative Combinatorics

Lecture Notes in Computer Science - Computer Algebra and Polynomials ◽

10.1007/978-3-319-15081-9_1 ◽

2015 ◽

pp. 1-29

Author(s):

Felix Breuer

Keyword(s):

Ehrhart Theory ◽

Enumerative Combinatorics ◽

Polyhedral Geometry

Download Full-text

Lattice Points and Simultaneous Core Partitions

The Electronic Journal of Combinatorics ◽

10.37236/5734 ◽

2018 ◽

Vol 25 (3) ◽

Cited By ~ 4

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.

Download Full-text

Quasisymmetric functions from combinatorial Hopf monoids and Ehrhart Theory

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.6334 ◽

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.

Download Full-text