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A floorplan is a tiling of a rectangle by rectangles. There are natural waysto order the elements---rectangles and segments---of a floorplan. Ackerman,Barequet and Pinter studied a pair of orders induced by neighborhood relationsbetween rectangles, and obtained a natural bijection between these pairs and(2-41-3, 3-14-2)-avoiding permutations, also known as (reduced) Baxterpermutations. In the present paper, we first perform a similar study for a pairof orders induced by neighborhood relations between segments of a floorplan. Weobtain a natural bijection between these pairs and another family ofpermutations, namely (2-14-3, 3-41-2)-avoiding permutations. Then, weinvestigate 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)-avoidingpermutations. In particular, we prove that the superposition of bothpermutations gives a complete Baxter permutation (originally calledw-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 Baxterpermutations. We enumerate these permutations. To our knowledge, thecharacterization of these permutations in terms of forbidden patterns and theirenumeration are both new results. Finally, we also study the special case ofthe so-called guillotine floorplans.

Orders induced by segments in floorplan partitions and (2-14-3,3-41-2)-avoiding permutations