Construct bijection between partitions of $[n-1]$ with $k-1$ parts, and partitions of $[n]$ with $k$ parts with no parts contains consecutive integers

Question

So... the title is just I want to ask for. This is exercise #104 of Chapter 1, in Lessons In Enumerative Combinatorics, GTM 290.

With previous exercise, I proved that number of $k-1$-part paritions of $[n-1]$(let me say this as set $X$), and $k$-part partitions of $[n]$ with no parts contains consecutive integers(let me say this as set $Y$).

But this exercise says construct bijection between $X$ and $Y$, so "They have same cardinalities so bijection exists" cannot be answer. I have seriously no idea to approach. Any help is welcome.

Edit

In this case, $[n]$ is abbreviate of $\{1,2,\cdots,n\}$, and $k$-partition of a set $S$ is mutually disjoint subset of $S$ $P_1,\cdots,P_k$ which satisfies $S=\bigcup_1^kP_i$.