On the Structure and Generators of the $n$th-order Chromatic Algebra
Ethan Yi-Heng Liu
TL;DR
The paper addresses the structure of the $n$th-order chromatic algebra $ ext{C}_n$ by providing a rigorous definition via an ideal $I_n$ and establishing a finite, constructive framework. It proves $ ext{dim}( ext{C}_n)=R_{2n}$, linking the algebra to Riordan numbers through a bijection between the chromatic-basis diagrams and noncrossing partitions without singletons, and introduces a quadratic generating set $E_n$ with $|E_n|=inom{n}{2}$ that generates $ ext{C}_n$, along with a method to build a basis from $E_n$. The work also refines the underlying relations to address edge-cases and clarifies the relationship to previous formulations (Fendley–Krushkal), while enabling an algebraic reformulation and prompting further questions about the kernel and generator degrees. Collectively, these results connect chromatic algebra to combinatorics and planar diagrammatics, enriching both the theoretical framework and potential applications to graph colorings and related invariants.
Abstract
This work investigates the intrinsic properties of the chromatic algebra, introduced by Fendley and Krushkal as a framework to study the chromatic polynomial. We prove that the dimension of the $n$th-order chromatic algebra is the $2n$th Riordan number, which exhibits exponential growth. We find a generating set of size $\binom{n}{2}$, and we provide a procedure to construct the basis from the generating set. We additionally provide proofs for fundamental facts about this algebra that appear to be missing from the literature. These include determining a representation of the chromatic algebra as noncrossing planar partitions and expanding the chromatic relations to include an edge case.