  A floorplan is a tiling of a rectangle by rectangles. There are natural ways
to order the elements---rectangles and segments---of a floorplan. Ackerman,
Barequet and Pinter studied a pair of orders induced by neighborhood relations
between rectangles, and obtained a natural bijection between these pairs and
(2-41-3, 3-14-2)-avoiding permutations, also known as (reduced) Baxter
permutations. In the present paper, we first perform a similar study for a pair
of orders induced by neighborhood relations between segments of a floorplan. We
obtain a natural bijection between these pairs and another family of
permutations, namely (2-14-3, 3-41-2)-avoiding permutations. Then, we
investigate relations between the two kinds of pairs of orders---and,
correspondingly, between (2-41-3, 3-14-2)- and (2-14-3, 3-41-2)-avoiding
permutations. In particular, we prove that the superposition of both
permutations gives a complete Baxter permutation (originally called
w-admissible, by Baxter and Joichi in the sixties). In other words, (2-14-3,
3-41-2)-avoiding permutations are the hidden part of complete Baxter
permutations. We enumerate these permutations. To our knowledge, the
characterization of these permutations in terms of forbidden patterns and their
enumeration are both new results. Finally, we also study the special case of
the so-called guillotine floorplans.
