Characters of the symmetric group corresponding to partitions into two parts Let $n\in\mathbb N$ be a natural number and $\lambda=(a,b)\vdash n$ a partition of $n$ into two parts, i.e. $a\ge b$ and $a+b=n$. In this special case, is there a simple description of the character $\chi_\lambda$ of the irreducible $S_n$-representation corresponding to $\lambda$? I have tried to deduce something from the Frobenius character formula and also using the Murnaghan-Nakayama recursion, but so far I couldn't really come up with a simple description. I would really appreciate any references/theorems in that direction.
 A: This is probably way too little too late, but someone just linked to this question from elsewhere on the site and seeing as it was never answered I thought I'd give an answer.
Consider the natural action of $S_n$ on ($\mathbb{C}$-linear combinations of) $b$-element subsets of $\{1,2,\dots,n\}$. This is not quite the representation you want but its character values are easy to compute.  For a given permutation the fixed subsets are exactly those which are unions of cycles of the permutation.  So for a permutation of cycle type $m_1, m_2, \dots$, (that is, it has $m_1$ $1$-cycles, $m_2$ $2$-cycles, etc.) the character value is the number of ways to write $b$ as a sum of at most $m_1$ ones, at most $m_2$ twos, etc.  This is just the coefficient of $x^b$ in $\Pi(1+x^i)^{m_i}$.
Now as I said, this isn't quite what you wanted, but it's close. Rather than being $\chi (n-b,b)$ this is $\chi (n-b,b) + \chi (n-b+1,b-1)+ \chi (n-b+2,b-2) + \dots \chi (n)$ (by say the Pieri rule).  In particular though you can take the answer you got above for $b$ and subtract the answer for $b-1$, and this will give you the irreducible character you are looking for.
