Reply to the referee's comments
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*** page 4 Def 2.1, the condition "there is exactly one rectangle" seems a
little artificial, what would go wrong if you replace "exactly" by "at
least", would the transitive closure be the same, would one miss an
important property (for instance, is the "exactly" necessary for Observation 2.4 to
hold?). To get more convinced of the necessity of the "exactly" it would be
good to comment on these points.

--- If one replaces "exactly one" by "at least one", the new relation
doesn't correspond anymore to the intuition of "neighbors", illustrated in
Fig. 4. In fact, we want the notion of neighbours to coincide with the covering
relation in the left-right order. Lemma 2.5 would be wrong if we would 
replace 'exactly one' by 'at least one':  this can be seen for instance 
in the example of Fig. 12, with the segments labelled 9, 10 and 11.

Having our notion of neighbors is for instance crucial in the constructions
of Section 3.4.

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*** page 11 line -4 Avoiding -> avoiding

--- Done.
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*** two remarks on Section 5:
* reduced Baxter permutations numbers are nicer when taken according to the
number of rises and descents.
Is it also possible to count 2-14-3,3-41-2-avoiding permutations according
to these two parameters ?

--- We tried this, but without any success. We added a remark about this.
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* did you try to obtain a generating tree for these permutations ? (which
could give another approach to count them)

--- Yes, and, indeed, the formula A(t) = 1/t B(t(1-t)) was first found and
proved in this way. But, once we found a combinatorial explanation (that
with improper pairs), we removed that proof.

We have now added a remark on this generating tree.
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- page 26 Proposition 6.5: separable-by-point permutations have an algebraic
specification (similar to the one for separable permutations),
which should give an alternative (more direct) computation of the generating
function. It would be good to show also this approach.

--- This is indeed possible, and we added a remark in this direction.

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- about the table page 27: another well known Baxter subfamily is the one
avoiding 2-41-3, 3-1-4-2, which correspond to non-separable maps.
It would be interesting to characterize the 2-14-3,3-41-2 permutations
associated to these permutations, similarly as you do for
separable permutations.

--- We added this as a suggestion for a new research. The paper is too long to
try to add something to it!

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