On the Structure and Generators of the $n$th-order Chromatic Algebra
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916088676089856 |
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| author | Liu, Ethan Yi-Heng |
| author_facet | Liu, Ethan Yi-Heng |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_06095 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Structure and Generators of the $n$th-order Chromatic Algebra Liu, Ethan Yi-Heng Combinatorics Rings and Algebras 05E16 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. |
| title | On the Structure and Generators of the $n$th-order Chromatic Algebra |
| topic | Combinatorics Rings and Algebras 05E16 |
| url | https://arxiv.org/abs/2401.06095 |