Annals of Combinatorics最新文献

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.

Recently, Blecher and Knopfmacher applied the notion of fixed points to integer partitions. This has already been generalized and refined in various ways such as h-fixed points for an integer parameter h by Hopkins and Sellers. Here, we consider the sequence of first column hook lengths in the Young diagram of a partition and corresponding fixed hooks. We enumerate these, using both generating function and combinatorial proofs, and find that they match occurrences of part sizes equal to their multiplicity. We establish connections to work of Andrews and Merca on truncations of the pentagonal number theorem and classes of partitions partially characterized by certain minimal excluded parts (mex).

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

We discuss sequential stub matching for directed graphs and show that this process can be used to sample simple digraphs with asymptotically equal probability. The process starts with an empty edge set and repeatedly adds edges to it with a certain state-dependent bias until the desired degree sequence is fulfilled, whilst avoiding placing a double edge or self-loop. We show that uniform sampling is achieved in the sparse regime when the maximum degree (d_text {max}) is asymptotically dominated by (m^{1/4}), where m is the number of edges. The proof is based on deriving various combinatorial estimates related to the number of digraphs with a given degree sequence and controlling concentration of these estimates in large digraphs. This suggests that sequential stub matching can be viewed as a practical algorithm for almost uniform sampling of digraphs. We show that this algorithm can be implemented to feature a linear expected runtime O(m).

A graph is cubical if it is a subgraph of a hypercube. For a cubical graph H and a hypercube (Q_n), (textrm{ex}(Q_n, H)) is the largest number of edges in an H-free subgraph of (Q_n). If (textrm{ex}(Q_n, H)) is at least a positive proportion of the number of edges in (Q_n), then H is said to have positive Turán density in the hypercube; otherwise it has zero Turán density. Determining (textrm{ex}(Q_n, H)) and even identifying whether H has positive or zero Turán density remains a widely open question for general H. In this paper we focus on layered graphs, i.e., graphs that are contained in an edge layer of some hypercube. Graphs H that are not layered have positive Turán density because one can form an H-free subgraph of (Q_n) consisting of edges of every other layer. For example, a 4-cycle is not layered and has positive Turán density. However, in general, it is not obvious what properties layered graphs have. We give a characterization of layered graphs in terms of edge-colorings. We show that most non-trivial subdivisions have zero Turán density, extending known results on zero Turán density of even cycles of length at least 12 and of length 8. However, we prove that there are cubical graphs of girth 8 that are not layered and thus having positive Turán density. The cycle of length 10 remains the only cycle for which it is not known whether its Turán density is positive or not. We prove that (textrm{ex}(Q_n, C_{10})= Omega (n2^n/ log ^a n)), for a constant a, showing that the extremal number for a 10-cycle behaves differently from any other cycle of zero Turán density.

We call a pair of vertex-disjoint, induced subtrees of a rooted tree twins if they have the same counts of vertices by out-degrees. The likely maximum size of twins in a uniformly random, rooted Cayley tree of size (nrightarrow infty ) is studied. It is shown that the expected number of twins of size ((2+delta )sqrt{log ncdot log log n}) approaches zero, while the expected number of twins of size ((2-delta )sqrt{log ncdot log log n}) approaches infinity.

The derived graph of a voltage graph consisting of a single vertex and two loops of different voltages is a circulant graph with two generators. We characterize the automorphism groups of connected, two-generator circulant graphs, and give their determining and distinguishing number, and when relevant, their cost of 2-distinguishing. We do the same for the subdivisions of connected, two-generator circulant graphs obtained by replacing one loop in the voltage graph with a directed cycle.

We calculate the volume of the tropical Prym variety of a harmonic double cover of metric graphs having non-trivial dilation. We show that the tropical Prym variety behaves discontinuously under deformations of the double cover that change the number of connected components of the dilation subgraph.