Partitions, or Young diagrams, index the representation of symmetric groups. There are plenty of combinatorial objects, constructed from partitions, that allow a better understanding of these representations. For instance: hooks, with the “hook formula” giving the dimensions of the representations in characteristic zero, and the notion of core, used in the theory of modular representations. The Plancherel measure on partitions has a natural meaning in representation theory, and since a theorem of Kerov–Vershik and Logan–Shepp we know that large partitions, chosen under the Plancherel measure, have a prescribed limit shape. One can then study the following problem: given a (combinatorial) quantity defined on partitions, study its behaviour as the size of the partition goes to infinity. For instance, Kerov–Vershik have studied the largest dimension of the representations, and recently we have studied the size of the cores. We will give other examples of quantities, defined on partitions, which should be interesting to study.