Is there a list of all set theoretic solutions $S:X \times X \to X \times X$ of the YBE for $X=\{1,2,3\}$? Or is it known how many solutions there are? I mean, $S_9$ is big but maybe not too big to check it with the computer?
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1 Answer
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2
Sure,
degenerate solutions:
\begin{equation} ((1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)), ((1, 1), (1, 2), (3, 1), (2, 1), (2, 2), (3, 2), (1, 3), (2, 3), (3, 3)), ((1, 1), (1, 2), (3, 2), (2, 1), (2, 2), (3, 1), (2, 3), (1, 3), (3, 3)), ((1, 1), (2, 1), (1, 3), (1, 2), (2, 2), (3, 2), (3, 1), (2, 3), (3, 3)), ((1, 1), (2, 1), (3, 1), (1, 2), (2, 2), (2, 3), (1, 3), (3, 2), (3, 3)), ((1, 1), (2, 3), (1, 3), (3, 2), (2, 2), (1, 2), (3, 1), (2, 1), (3, 3)), ((1, 1), (3, 1), (2, 1), (1, 3), (2, 2), (2, 3), (1, 2), (3, 2), (3, 3)) \end{equation}
non degenerate solutions:
\begin{equation} ((1, 1), (2, 1), (3, 1), (1, 2), (2, 2), (3, 2), (1, 3), (2, 3), (3, 3)), ((1, 1), (2, 1), (3, 1), (1, 2), (2, 2), (3, 2), (2, 3), (1, 3), (3, 3)), ((1, 1), (2, 1), (3, 1), (1, 2), (2, 3), (3, 3), (1, 3), (2, 2), (3, 2)), ((1, 1), (2, 1), (3, 1), (1, 2), (3, 2), (2, 2), (1, 3), (3, 3), (2, 3)), ((1, 1), (2, 1), (3, 1), (1, 2), (3, 3), (2, 3), (1, 3), (3, 2), (2, 2)), ((1, 1), (2, 1), (3, 1), (1, 3), (2, 2), (3, 2), (1, 2), (2, 3), (3, 3)), ((1, 1), (2, 1), (3, 1), (1, 3), (2, 3), (3, 3), (1, 2), (2, 2), (3, 2)), ((1, 1), (2, 1), (3, 1), (1, 3), (3, 2), (2, 2), (1, 2), (3, 3), (2, 3)), ((1, 1), (2, 1), (3, 1), (1, 3), (3, 3), (2, 3), (1, 2), (3, 2), (2, 2)), ((1, 1), (2, 1), (3, 1), (3, 2), (2, 2), (1, 2), (1, 3), (2, 3), (3, 3)), ((1, 1), (2, 1), (3, 1), (3, 3), (1, 3), (2, 3), (2, 2), (3, 2), (1, 2)), ((1, 1), (2, 1), (3, 2), (1, 2), (2, 2), (3, 1), (1, 3), (2, 3), (3, 3)), ((1, 1), (2, 1), (3, 2), (1, 2), (2, 2), (3, 1), (2, 3), (1, 3), (3, 3)), ((1, 1), (2, 3), (3, 1), (1, 2), (2, 2), (3, 2), (1, 3), (2, 1), (3, 3)), ((1, 1), (2, 3), (3, 1), (3, 2), (2, 2), (1, 2), (1, 3), (2, 1), (3, 3)), ((1, 1), (2, 3), (3, 2), (1, 3), (2, 2), (3, 1), (1, 2), (2, 1), (3, 3)), ((1, 1), (3, 1), (2, 1), (1, 2), (2, 2), (3, 2), (1, 3), (2, 3), (3, 3)), ((1, 1), (3, 1), (2, 1), (1, 2), (2, 3), (3, 3), (1, 3), (2, 2), (3, 2)), ((1, 1), (3, 1), (2, 1), (1, 2), (3, 2), (2, 2), (1, 3), (3, 3), (2, 3)), ((1, 1), (3, 1), (2, 1), (1, 2), (3, 3), (2, 3), (1, 3), (3, 2), (2, 2)), ((1, 1), (3, 1), (2, 1), (1, 3), (2, 2), (3, 2), (1, 2), (2, 3), (3, 3)), ((1, 1), (3, 1), (2, 1), (1, 3), (2, 3), (3, 3), (1, 2), (2, 2), (3, 2)), ((1, 1), (3, 1), (2, 1), (1, 3), (3, 2), (2, 2), (1, 2), (3, 3), (2, 3)), ((1, 1), (3, 1), (2, 1), (1, 3), (3, 3), (2, 3), (1, 2), (3, 2), (2, 2)), ((1, 1), (3, 1), (2, 1), (3, 2), (2, 2), (1, 2), (2, 3), (1, 3), (3, 3)), ((1, 1), (3, 3), (2, 2), (1, 2), (3, 1), (2, 3), (1, 3), (3, 2), (2, 1)), ((1, 2), (2, 2), (3, 1), (1, 1), (2, 1), (3, 2), (1, 3), (2, 3), (3, 3)), ((1, 2), (2, 2), (3, 1), (1, 1), (2, 1), (3, 2), (2, 3), (1, 3), (3, 3)), ((1, 2), (2, 2), (3, 2), (1, 1), (2, 1), (3, 1), (1, 3), (2, 3), (3, 3)), ((1, 2), (2, 2), (3, 2), (1, 1), (2, 1), (3, 1), (2, 3), (1, 3), (3, 3)), ((1, 2), (2, 2), (3, 2), (1, 3), (2, 3), (3, 3), (1, 1), (2, 1), (3, 1)), ((1, 3), (2, 1), (3, 3), (1, 2), (2, 2), (3, 2), (1, 1), (2, 3), (3, 1)), ((1, 3), (2, 1), (3, 3), (3, 2), (2, 2), (1, 2), (1, 1), (2, 3), (3, 1)), ((1, 3), (2, 3), (3, 3), (1, 1), (2, 1), (3, 1), (1, 2), (2, 2), (3, 2)), ((1, 3), (2, 3), (3, 3), (1, 2), (2, 2), (3, 2), (1, 1), (2, 1), (3, 1)), ((1, 3), (2, 3), (3, 3), (3, 2), (2, 2), (1, 2), (1, 1), (2, 1), (3, 1)), ((2, 1), (1, 1), (3, 1), (2, 2), (1, 2), (3, 2), (1, 3), (2, 3), (3, 3)), ((2, 1), (1, 1), (3, 1), (2, 2), (1, 2), (3, 2), (2, 3), (1, 3), (3, 3)), ((2, 1), (1, 1), (3, 2), (2, 2), (1, 2), (3, 1), (1, 3), (2, 3), (3, 3)), ((2, 1), (1, 1), (3, 2), (2, 2), (1, 2), (3, 1), (2, 3), (1, 3), (3, 3)), ((2, 1), (3, 1), (1, 1), (2, 2), (3, 2), (1, 2), (2, 3), (3, 3), (1, 3)), ((2, 2), (1, 2), (3, 1), (2, 1), (1, 1), (3, 2), (1, 3), (2, 3), (3, 3)), ((2, 2), (1, 2), (3, 1), (2, 1), (1, 1), (3, 2), (2, 3), (1, 3), (3, 3)), ((2, 2), (1, 2), (3, 2), (1, 3), (3, 3), (2, 3), (3, 1), (2, 1), (1, 1)), ((2, 2), (1, 2), (3, 2), (2, 1), (1, 1), (3, 1), (1, 3), (2, 3), (3, 3)), ((2, 2), (1, 2), (3, 2), (2, 1), (1, 1), (3, 1), (2, 3), (1, 3), (3, 3)), ((2, 2), (3, 1), (1, 3), (2, 1), (3, 3), (1, 2), (2, 3), (3, 2), (1, 1)), ((2, 2), (3, 2), (1, 2), (2, 3), (3, 3), (1, 3), (2, 1), (3, 1), (1, 1)), ((2, 3), (1, 2), (3, 1), (2, 1), (1, 3), (3, 2), (2, 2), (1, 1), (3, 3)), ((2, 3), (3, 3), (1, 3), (1, 2), (2, 2), (3, 2), (3, 1), (1, 1), (2, 1)), ((2, 3), (3, 3), (1, 3), (2, 1), (3, 1), (1, 1), (2, 2), (3, 2), (1, 2)), ((3, 1), (1, 1), (2, 1), (3, 2), (1, 2), (2, 2), (3, 3), (1, 3), (2, 3)), ((3, 1), (2, 1), (1, 1), (1, 2), (2, 2), (3, 2), (3, 3), (2, 3), (1, 3)), ((3, 1), (2, 1), (1, 1), (3, 2), (2, 2), (1, 2), (3, 3), (2, 3), (1, 3)), ((3, 1), (2, 3), (1, 1), (1, 2), (2, 2), (3, 2), (3, 3), (2, 1), (1, 3)), ((3, 1), (2, 3), (1, 1), (3, 2), (2, 2), (1, 2), (3, 3), (2, 1), (1, 3)), ((3, 2), (1, 2), (2, 2), (2, 1), (3, 1), (1, 1), (1, 3), (2, 3), (3, 3)), ((3, 2), (1, 2), (2, 2), (3, 3), (1, 3), (2, 3), (3, 1), (1, 1), (2, 1)), ((3, 2), (2, 1), (1, 3), (3, 3), (2, 2), (1, 1), (3, 1), (2, 3), (1, 2)), ((3, 3), (1, 2), (2, 1), (3, 2), (1, 1), (2, 3), (3, 1), (1, 3), (2, 2)), ((3, 3), (1, 3), (2, 3), (3, 1), (1, 1), (2, 1), (3, 2), (1, 2), (2, 2)), ((3, 3), (2, 1), (1, 3), (1, 2), (2, 2), (3, 2), (3, 1), (2, 3), (1, 1)), ((3, 3), (2, 1), (1, 3), (3, 2), (2, 2), (1, 2), (3, 1), (2, 3), (1, 1)), ((3, 3), (2, 3), (1, 3), (1, 2), (2, 2), (3, 2), (3, 1), (2, 1), (1, 1)), ((3, 3), (2, 3), (1, 3), (2, 1), (1, 1), (3, 1), (1, 2), (3, 2), (2, 2)), ((3, 3), (2, 3), (1, 3), (3, 2), (2, 2), (1, 2), (3, 1), (2, 1), (1, 1)) \end{equation}
Best Regards, Pablo
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4$\begingroup$ Thanks for sharing ! how to read these notations ? $\endgroup$ Commented Feb 4 at 19:02
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$\begingroup$ each permutation is the image of $X\times X$ in the lexicographical order. $\endgroup$– parmasCommented Feb 4 at 19:09