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

TL;DR

This paper explores the relationship between segment-induced orders in floorplan partitions and certain pattern-avoiding permutations, establishing bijections, characterizations, and enumeration results for these combinatorial structures.

Contribution

It introduces a new bijection between segment-induced order pairs and (2-14-3, 3-41-2)-avoiding permutations, and characterizes these permutations with forbidden patterns, including enumeration.

Findings

Bijection between segment orders and (2-14-3, 3-41-2)-avoiding permutations

New characterization of these permutations via forbidden patterns

Enumeration of the permutations and analysis of special floorplan cases

Abstract

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.