Combinatorics of partitions and tableaux

In Chapter 2, we introduced the notion of integer partition, and we saw that integer partitions labeled the bases of the algebra Sym, and the irreducible representations of the symmetric groups. Our goal is now to use these combinatorial objects in order to compute quantities such as the dimensions of the modules S^λ , or their characters on specific conjugacy classes. In this framework, a central role is played by chains of partitions λ ⁽¹⁾ ⊂ λ ⁽²⁾ ⊂ ⋯ ⊂ λ ^(r). These chains are also called tableaux, and they can be interpreted as numberings of the cells of the Young diagram of the final partition λ ^(r), with some condition of growth of the numbers along the rows and the columns of the Young diagram.