This thesis explores higher-dimensional generalizations of classical combinatorial objects such as permutations, combinatorial maps, and mosaic floorplans. Its comprehensive goal is to develop combinatorial tools to better understand these higher dimensional objects.
The first part of this thesis focuses on the combinatorial properties of random tensor models, whose Feynman graphs are (d + 1)-colored graphs that extend the notion of combinatorial maps. More precisely, we are interested here in two asymptotic expansions of tensor models: the 1/N (where N is the size of the tensors) expansion and the double-scaling limit. We also investiguate some duality properties of tensor models with orthogonal and symplectic symmetries.
The second part of this thesis focuses on d-floorplans and d-permutations, the higher dimensional analogs of mosaic floorplans and permutations. After some reminders on the 2-dimensional case, we construct a generating tree for d-floorplans. The structure of this tree generalizes in a non-trivial way the one of mosaic floorplans. Then, we establish a bijection between (2^{d-1})-floorplans and d-permutations characterized by forbidden patterns. This bijection generalizes the one between mosaic floorplans and Baxter permutations.