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Though your interests are partly separate from the purely mathematical framework here, the basic theory is well developed. Notation and terminology vary, of course: e.g., your "semi-ordering" is usually called a "partial ordering". In the case of the symmetric group $S_n$, a convenient modern treatment is given by Gordon James in The Representation Theory ...


Let $\gamma(x) = (\alpha(x), \beta(x)),$ where $\alpha, \beta$ are the real and imaginary parts. A self-intersection corresponds to a simultaneous zero of $(\alpha(x)-\alpha(y))/(x-y)$ and $(\beta(x)-\beta(y)/(x-y),$ If you use the rational parametrization for the circle (the $\tan t/2$ trick), both expressions become polynomials, and the number of ...

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