If $G$ acts transitively on $X$, then the permutation representation of $G$ is induced from the permutation representation of an arbitrary the stabilizer $S$. S$of an arbitrary point. As a result,$\langle \chi , \chi \rangle$counts the number of orbits of$S$on$X$. In your situation$G=S_n$is acting transitively on$X=\{1,\ldots,n\}$. Let$S$be the stabilizer of$n$; this is pretty much$S_{n-1} \subset S_n$, which has two orbits on$X$:$\{1,\ldots, n-1\}$and$\{n\}$. 1 Here's another proof. If$G$acts transitively on$X$, then the permutation representation of$G$is induced from the permutation representation of an arbitrary stabilizer$S$. As a result,$\langle \chi , \chi \rangle$counts the number of orbits of$S$on$X$. In your situation$G=S_n$is acting transitively on$X=\{1,\ldots,n\}$. Let$S$be the stabilizer of$n$; this is pretty much$S_{n-1} \subset S_n$, which has two orbits on$X$:$\{1,\ldots, n-1\}$and$\{n\}\$.