This might be a classic question, but since I am new to representation theory of the symmetric group, I am asking it here.


Suppose that $\lambda\_1 \geq \lambda\_2 \geq \dots \lambda\_k$ and $\rho$ be the irreducible representation of $S\_n$ associated with Young tableau $(\lambda\_1,\dots,\lambda\_k)$ where $n=\sum \lambda\_i$. What can be said about dimension of $\rho$? I am specifically interested in the case where we have two rows (i.e. $k=2$) and the case where $\lambda\_i =1$ for $i\geq 3$.



Dimension of irreducible representation associated to a Young tableau
This is a classical application of Frobenius formula and Vandermonde determinant. We have $$\dim V\_{\lambda}=\frac{n!}{l\_1! \cdots l\_k!} \prod\_{i<j}(l\_i-l\_j),$$
where $l\_i=\lambda\_i+k-i$. See Section 4.1 of 


W. Fulton, J. Harris: *Representation Theory* (*a first course*), GTM **129** (1991). 


