2025/04/01 by Bonichon, Nicolas, Muller, Thomas, Tanasa, Adrian
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2504.01116
A 2-dimensional mosaic floorplan is a partition of a rectangle by other rectangles with no empty rooms. These partitions (considered up to some deformations) are known to be in bijection with Baxter permutations. A d-floorplan is the generalisation of mosaic floorplans in higher dimensions, and a d-permutation is a (d-1)-tuple of permutations. Recently, in N. Bonichon and P.-J. Morel, \it J. Integer Sequences 25 (2022), Baxter d-permutations generalising the usual Baxter permutations were introduced. In this paper, we consider mosaic floorplans in arbitrary dimensions, and we construct a generating tree for d-floorplans, which generalises the known generating tree structure for 2-floorplans. The corresponding labels and rewriting rules appear to be significantly more involved in higher dimensions. Moreover we give a bijection between the 2d-1-floorplans and d-permutations characterized by forbidden vincular patterns. Surprisingly, this set of d-permutations is strictly contained within the set of Baxter d-permutations.