Spencer Backman
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View article: Skeletal generalizations of Dyck paths, parking functions, and chip-firing games
Skeletal generalizations of Dyck paths, parking functions, and chip-firing games Open
For $0\leq k\leq n-1$, we introduce a family of $k$-skeletal paths which are counted by the $n$-th Catalan number for each $k$, and specialize to Dyck paths when $k=n-1$. We similarly introduce $k$-skeletal parking functions which are equi…
View article: Convex Geometry of Building Sets
Convex Geometry of Building Sets Open
Building sets were introduced in the study of wonderful compactifications of hyperplane arrangement complements and were later generalized to finite meet-semilattices. Convex geometries, the duals of antimatroids, offer a robust combinator…
View article: A Regular Unimodular Triangulation of the Matroid Base Polytope
A Regular Unimodular Triangulation of the Matroid Base Polytope Open
We produce the first regular unimodular triangulation of an arbitrary matroid base polytope. We then extend our triangulation to integral generalized permutahedra. Prior to this work it was unknown whether each matroid base polytope admitt…
View article: Simplicial generation of Chow rings of matroids
Simplicial generation of Chow rings of matroids Open
We introduce a presentation of the Chow ring of a matroid by a new set of generators, called “simplicial generators.” These generators are analogous to nef divisors on projective toric varieties, and admit a combinatorial interpretation vi…
View article: Fan Valuations and spherical intrinsic volumes
Fan Valuations and spherical intrinsic volumes Open
We generalize valuations on polyhedral cones to valuations on fans. For fans induced by hyperplane arrangements, we show a correspondence between rotation-invariant valuations and deletion-restriction invariants. In particular, we define a…
View article: Matroid Chern-Schwartz-MacPherson cycles and Tutte activities
Matroid Chern-Schwartz-MacPherson cycles and Tutte activities Open
Lopéz de Medrano-Rinćon-Shaw defined Chern-Schwartz-MacPherson cycles for an arbitrary matroid $M$ and proved by an inductive geometric argument that the unsigned degrees of these cycles agree with the coefficients of $T(M;x,0)$, where $T(…
View article: Fourientation activities and the Tutte polynomial
Fourientation activities and the Tutte polynomial Open
A fourientation of a graph G is a choice for each edge of the graph whether to orient that edge in either direction, leave it unoriented, or biorient it. We may naturally view fourientations as a mixture of subgraphs and graph orientations…
View article: Tutte Polynomial Activities
Tutte Polynomial Activities Open
Unlike Whitney's definition of the corank-nullity generating function $T(G;x+1,y+1)$, Tutte's definition of his now eponymous polynomial $T(G;x,y)$ requires a total order on the edges of which the polynomial is a posteriori independent. Tu…
View article: Simplicial generation of Chow rings of matroids
Simplicial generation of Chow rings of matroids Open
We introduce a presentation of the Chow ring of a matroid by a new set of generators, called "simplicial generators." These generators are analogous to nef divisors on projective toric varieties, and admit a combinatorial interpretation vi…
View article: Topological Bijections for Oriented Matroids
Topological Bijections for Oriented Matroids Open
In previous work by the first and third author with Matthew Baker, a family of bijections between bases of a regular matroid and the Jacobian group of the matroid was given. The core of the work is a geometric construction using zonotopal …
View article: Extension-lifting Bijections for Oriented Matroids
Extension-lifting Bijections for Oriented Matroids Open
Extending the notion of geometric bijections for regular matroids, introduced by the first and third author with Matthew Baker, we describe a family of bijections between bases of an oriented matroid and special orientations. These bijecti…
View article: GEOMETRIC BIJECTIONS FOR REGULAR MATROIDS, ZONOTOPES, AND EHRHART THEORY
GEOMETRIC BIJECTIONS FOR REGULAR MATROIDS, ZONOTOPES, AND EHRHART THEORY Open
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 cri…
View article: Cone valuations, Gram's relation, and flag-angles
Cone valuations, Gram's relation, and flag-angles Open
Interior angle vectors of polytopes are semi-discrete analogs of $f$-vectors that take into account the interior angles at faces measured by spherical volumes. In this context, Gram's relation takes the place of the Euler-Poincare relation…
View article: Generalized angle vectors, geometric lattices, and flag-angles
Generalized angle vectors, geometric lattices, and flag-angles Open
Interior and exterior angle vectors of polytopes capture curvature information at faces of all dimensions and can be seen as metric variants of $f$-vectors. In this context, Gram's relation takes the place of the Euler--Poincaré relation a…
View article: Transfinite Ford–Fulkerson on a finite network
Transfinite Ford–Fulkerson on a finite network Open
It is well-known that the Ford–Fulkerson algorithm for finding a maximum flow in a network need not terminate if we allow the arc capacities to take irrational values. Every non-terminating example converges to a limit flow, but this limit…
View article: A convolution formula for Tutte polynomials of arithmetic matroids and other combinatorial structures
A convolution formula for Tutte polynomials of arithmetic matroids and other combinatorial structures Open
In this note we generalize the convolution formula for the Tutte polynomial of Kook-Reiner-Stanton and Etienne-Las Vergnas to a more general setting that includes both arithmetic matroids and delta-matroids. As corollaries, we obtain new p…