We call the four-colored noncrossing partition representation induced by the canonical sequence of separating points the canonical representation of a.
Proposition 3.3 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.
For example, in the Human variant X1 of CSF2RB (XM_005261340) the intron upstream of exon 7 ends by an "early" CCTCAG while the corresponding intron in the canonical sequence
(NM_007780) ends at another CCTCAG 18 nt downstream, resulting in an 18-nt longer exon in the variant.
In this section we define a family of operation sequences, called canonical, such that each operation sequence is equivalent to exactly one canonical sequence. Canonical sequences are thus in one-to-one correspondence with permutations that can be produced with two parallel stacks.
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.