Appears in collection : Combinatorics and Arithmetic for Physics

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 $d$-permutation is a $(d-1)$-tuple of permutations. Recently, in N. Bonichon and P.-J. Morel, J. Integer Sequences 25 (2022), Baxter $d$-permutations generalising the usual Baxter permutations were introduced. In this talk, I will introduce the $d$-floorplans which generalise the mosaic floorplans to arbitrary dimensions. Then, I will present the construction of their generating tree. The corresponding labels and rewriting rules appear to be significantly more involved in higher dimensions. Finally, I will present a bijection between the $2^{d-1}$-floorplans and $d$-permutations characterised by forbidden vincular patterns. Surprisingly, this set of $d$-permutations is strictly contained within the set of Baxter $d$-permutations. This is a joint work with Nicolas Bonichon and Adrian Tanasa (Université de bordeaux), this talk is based on arXiv:2504.01116.