Show that any number of partitions of $r+k$ into $k$ parts is equal to the number of partitions of $$r+\binom{k+1}{2}$$ into $k$ distinct parts for $r \geq k$.
I would like to see a proof for this, my try was incorrect and gives no result. Does it work for $k \geq r$?
HINT: The Ferrers diagram of any partition of $r+\binom{k+1}2=r+\sum_{i=1}^ki$ into $k$ distinct parts must have at least $1$ dot in the bottom row, at least $2$ dots, in the next row up, and so on, all the way up to at least $k$ dots in the top row. What do you get if leave the bottom row alone, remove $1$ dot from the next row up, remove $2$ dots from the row above that, and so on, finally removing $k-1$ dots from the top row?