Relative volume of comparable pairs under semigroup majorization

Any semigroup \(\mathcal {S}\) of stochastic matrices induces a semigroup majorization relation \(\prec ^{\mathcal {S}}\) on the set \(\Delta _{n-1}\) of probability n-vectors. Pick X, Y at random in \(\Delta _{n-1}\): what is the probability that X and Y are comparable under \(\prec ^{\mathcal {S}}\)? We review recent asymptotic (\(n\rightarrow \infty \)) results and conjectures in the case of majorization relation (when \(\mathcal {S}\) is the set of doubly stochastic matrices), discuss natural generalisations, and prove a new asymptotic result in the case of majorization, and new exact finite-n formulae in the case of UT-majorization relation, i.e. when \(\mathcal {S}\) is the set of upper-triangular stochastic matrices.