Bipartite solutions of the set-theoretic Yang--Baxer equation

@misc{militaru2026bipartite,
  title = {{Bipartite solutions of the set-theoretic Yang--Baxer equation}},
  author = {Gigel Militaru},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.30845},
  eprint = {2610.30845},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.30845},
}

TY  - UNPB
TI  - Bipartite solutions of the set-theoretic Yang--Baxer equation
AU  - Gigel Militaru
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2610.30845
UR  - https://arxiv.org/abs/2610.30845
AB  - We introduce a new class of unitary set-theoretic Yang--Baxter solutions, called \emph{bipartite} solutions. This class, which is purely combinatorial in nature, forms a full subcategory of ℒ𝒴ℬ, the category of Lyubashenko type solutions of Agore, Chirvasitu and the author, and is particularly amenable to explicit classification results. We give an intrinsic necessary and sufficient condition for a solution of the Yang--Baxter equation to be isomorphic to a bipartite solution, we settle the isomorphism problem within this class and, as an application, we determine their automorphism groups. If X is a finite set with |X| = n, the number of isomorphism classes of bipartite solutions on X is ⌊ n/2 ⌋ + 1 + ∑_m=2^n ( o(m, n−m) − 1 ), where $o(m, k)$ denotes the number of orbits of the symmetric group S_m acting by conjugation on the k-element multisets of elements of S_m. The paper also offers a comparative study: for small |X| = n we count, up to isomorphism, all set-theoretic Yang--Baxter solutions on X, together with five distinguished families of solutions. Throughout the range we were able to compute, the class ℒ𝒴ℬ accounts for a large share of all bijective and of all non-degenerate solutions, strikingly close to the total.
ER  -