The Cloud and Flock Polynomials of q-Matroids

2025-01-24

AI Summary

This paper investigates structural relationships between the Whitney function and the cloud and flock polynomials of $q$-matroids. Using cyclic flat lattice analysis, Whitney function decomposition, and algebraic combinatorial techniques, we establish for the first time that both the Whitney function and all cloud/flock polynomials are fully determined by the $q$-matroid’s configuration. Moreover, these polynomials provide a complete characterization of the Whitney function and depend solely on the structure of the cyclic flat lattice together with rank–nullity data. We further show the converse does not hold: distinct configurations can yield identical cloud or flock polynomials, and we construct explicit counterexamples. Additionally, we demonstrate that the Whitney function of a direct sum $q$-matroid is not uniquely determined by the Whitney functions of its components—revealing an intrinsic limitation of these invariants under combinatorial composition.

Technology Category

Knowledge Representation and Reasoning: Qualitative ReasoningConstraint Satisfaction and Optimization: Other Foundations of Constraint SatisfactionSearch and Optimization: Combinatorial Optimization

Application Category

Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphsWeb Mining and Content Analysis: Models for Web evolutionSystems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

We show that the Whitney function of a q-matroid can be determined from the cloud and flock polynomials associated to the cyclic flats. These polynomials capture information about the corank (resp., nullity) of certain spaces whose cyclic core (resp., closure) is the given cyclic flat. Going one step further, we prove that the Whitney function, and in fact all cloud and flock polynomials, are determined by the configuration of the q-matroid, that is the abstract lattice of cyclic flats together with the corank-nullity data. Examples illustrate that the converses of the above statements are not true. This has the consequence that the Whitney function of a direct sum is not determined by the Whitney functions of the summands.

Problem

• q-polymatroids
• Whitney functions
• polynomial representations

Innovation

• q-analogs
• Whitney functions
• Polynomial methods