SupplementaryMaterialfor " Permuton-induced ChineseRestaurantProcess "

As shown by the existing BNP models such as BBP [18], RTP [19], and MP [22, 21], the most important rectangular partitioning classes in terms of applications in relational data analysis are hierarchical partitionings andgeneric rectangulations. Unfortunately,tothebestofourknowledge, however,the permutation class with the one-to-one correspondence with hierarchical partitioning are still under development, and we do not knowhowto handle uniform hierarchical partitioning explicitly throughpermuton. However,tothebestofour knowledge, there is still no way to construct the corresponding permuton to uniform2-clumped permutations. Finally, using the Lebesgue preserving functionϕ: [0,1] [0,1] such that (a,b) [0,1] is an inversion if and only if the sign ofmint [a,b]e(t) (i.e., F(mint [a,b]e(t))) is, we obtain the Brownianseparablepermuton: γ() = Leb t [0,1];(t,ϕ(t)) . The construction of the Baxter permuton is quite complicated, involving bipolar orientations and walks in the quadrant, therefore we only sketch it here.