Annals of Combinatorics最新文献

The inquiry into identifying sets of monomials that can be eliminated from a generic homogeneous polynomial via a linear change of coordinates was initiated by E. K. Wakeford. This linear algebra problem prompted C. K. Fan and J. Losonczy to introduce the notion of acyclic matchings in the additive group (mathbb {Z}^n), subsequently extended to abelian groups by the latter author. Alon, Fan, Kleitman, and Losonczy established the acyclic matching property for (mathbb {Z}^n). This note aims to classify all abelian groups with respect to the acyclic matching property.

We consider the restrictions of Shi arrangements to Weyl cones, their relations to antichains in the root poset, and their intersection posets. For any Weyl cone, we provide bijections between regions, flats intersecting the cone, and antichains of a naturally defined subposet of the root poset. This gives a refinement of the parking function numbers via the Poincaré polynomials of the intersection posets of all Weyl cones. Finally, we interpret these Poincaré polynomials as the Hilbert series of three isomorphic graded rings. One of these rings arises from the Varchenko–Gel’fand ring, another is the coordinate ring of the vertices of the order polytope of a subposet of the root poset, and the third is purely combinatorial.

An orientable vertex primitive complete map is a two-cell embedding of a complete graph into an orientable surface such that the automorphism group of this map acts primitively on its vertex set. The paper is devoted to the problem of enumerating orientable vertex primitive complete maps. For a given integer n, we derive the number of different such maps with n vertices. Furthermore, we obtain explicit formulas for the numbers of non-isomorphic orientable vertex primitive complete maps with n vertices.

We analyze polyhedra composed of hexagons and triangles with three faces around each vertex, and their 3-regular planar graphs of edges and vertices, which we call “trihexes”. Trihexes are analogous to fullerenes, which are 3-regular planar graphs whose faces are all hexagons and pentagons. Every trihex can be represented as the quotient of a hexagonal tiling of the plane under a group of isometries generated by (180^circ ) rotations. Every trihex can also be described with either one or three “signatures”: triples of numbers that describe the arrangement of the rotocenters of these rotations. Simple arithmetic rules relate the three signatures that describe the same trihex. We obtain a bijection between trihexes and equivalence classes of signatures as defined by these rules. Labeling trihexes with signatures allows us to put bounds on the number of trihexes for a given number of vertices v in terms of the prime factorization of v and to prove a conjecture concerning trihexes that have no “belts” of hexagons.

We prove a new determinantal formula for the characters of irreducible representations of orthosymplectic Lie superalgebras analogous to the formula developed by Moens and Jeugt (J Algebraic Combin 17(3):283–307, 2003) for general linear Lie superalgebras. Our proof uses the Jacobi–Trudi type formulas for orthosymplectic characters. As a consequence, we show that the odd symplectic characters introduced by Proctor (Invent Math 92(2):307–332, 1988) are the same as the orthosymplectic characters with some specialized indeterminates. We also give a generalization of an odd symplectic character identity due to Brent, Krattenthaler and Warnaar (J Combin Theory Ser A 144:80–138, 2016).

Efficient deterministic algorithms to construct representations of lattice path matroids over finite fields are presented. They are built on known constructions of hierarchical secret sharing schemes, a recent characterization of hierarchical matroid ports, and the existence of isolating weight functions for lattice path matroids whose values are polynomial on the size of the ground set.

We prove that the acyclic reorientation poset of a directed acyclic graph D is a lattice if and only if the transitive reduction of any induced subgraph of D is a forest. We then show that the acyclic reorientation lattice is always congruence normal, semidistributive (thus congruence uniform) if and only if D is filled, and distributive if and only if D is a forest. When the acyclic reorientation lattice is semidistributive, we introduce the ropes of D that encode the join irreducible acyclic reorientations and exploit this combinatorial model in three directions. First, we describe the canonical join and meet representations of acyclic reorientations in terms of non-crossing rope diagrams. Second, we describe the congruences of the acyclic reorientation lattice in terms of lower ideals of a natural subrope order. Third, we use Minkowski sums of shard polytopes of ropes to construct a quotientope for any congruence of the acyclic reorientation lattice.

Grothendieck polynomials (mathfrak {G}_w) of permutations (win S_n) were introduced by Lascoux and Schützenberger (C R Acad Sci Paris Sér I Math 295(11):629–633, 1982) as a set of distinguished representatives for the K-theoretic classes of Schubert cycles in the K-theory of the flag variety of (mathbb {C}^n). We conjecture that the exponents of nonzero terms of the Grothendieck polynomial (mathfrak {G}_w) form a poset under componentwise comparison that is isomorphic to an induced subposet of (mathbb {Z}^n). When (win S_n) avoids a certain set of patterns, we conjecturally connect the coefficients of (mathfrak {G}_w) with the Möbius function values of the aforementioned poset with (hat{0}) appended. We prove special cases of our conjectures for Grassmannian and fireworks permutations