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We call the four-colored noncrossing partition representation induced by the canonical sequence of separating points the canonical representation of a.
3 Every reduced permutation of genus 1 of has a unique canonical sequence (a, b, c, d) of separating points, that induces a four-colored noncrossing partition representation.
4 Let [alpha] = [PHI]([beta], [gamma]) be the representation of the reduced permutation a of genus 1 induced by its canonical sequence of separating points (a, b, c, d).
In Section 3, we establish a system of functional equations that characterises the generating function of canonical sequences, and thus, of permutations that can be produced by two parallel stacks.