# The construction of multipermutation solutions of the Yang-Baxter equation of level 2

## TL;DR

This paper classifies and constructs involutive set-theoretic solutions to the Yang-Baxter equation of level 2, distinguishing between distributive and non-distributive types, and provides enumeration results up to size 14.

## Contribution

It introduces a new construction method for distributive solutions using abelian groups and matrices, and classifies solutions into two classes with enumeration up to size 14.

## Key findings

- Distributive solutions can be constructed from abelian groups and matrices.
- Non-distributive solutions can be derived from distributive solutions and permutations.
- All distributive involutive solutions up to size 14 are enumerated.

## Abstract

We study involutive set-theoretic solutions of the Yang-Baxter equation of multipermutation level 2. These solutions happen to fall into two classes -- distributive ones and non-distributive ones. The distributive ones can be effectively constructed using a set of abelian groups and a matrix of constants. Using this construction, we enumerate all distributive involutive solutions up to size 14. The non-distributive solutions can be also easily constructed, using a distributive solution and a permutation.