Show that for any positive integer $n>1$, the number of partitions of $n$ in which the two largest parts are equal is $p(n) − p(n − 1)$.
I know that by a theorem in my textbook that $p(n) − p(n − 1)$ is equal to the number of partitions of n in which each part is at least 2. I have figured out that the number of partitions of n in which each part is at least 2 is the conjugate of the two largest parts being equal but I am not sure why. I will complete my proof by stating that they are conjugates and therefore equal but I want to understand why.
If the two largest parts of a partition are not equal, say $\lambda_1>\lambda_2$, the conjugate partition has $\lambda_1-\lambda_2\ge 1$ parts of size $1$. So, by contraposition, smallest part at least $2$ implies that the conjugate has its two largest parts equal.