Representation Theory of Symmetric Groups and Combinatorics

This paper contains an introduction of representation theory of symmetric groups and several applications in Combinatorics. By using Hook length Formula, we then consider two specific shaped Standard Young Tableaux (SYT) where (n, n) shaped SYT has a connection with Catalan Numbers and the number of ways to fill a triangular formed SYT represents the largest irreducible representation of such symmetric group.