Construct a bijection between unrestricted partitions of $[n − 1]$ into $k − 1$ parts and partitions of $[n]$ into $k$ parts in which no part contains consecutive integers.
I was thinking of picking a partition of $[n-1]$ into $k-1$ parts $\{A_1,A_2,\dots,A_{k-1}\}$ and first look to the block to which belongs $n-1$, then if $n-2$ belonged to this same block I'd remove it and add it to a new set $A'_{k}$ that will be my $k$-th part in the partition of $[n]$ into $k$ parts. Then, if needed, I'd repeat the same process for the block that contains $n-1$, if $a$ is now the second greatest element in this block and $a-1$ also belongs to this same part, I'd remove $a-1$ and add it to the set $A'_{k}$. After repeating this process for every block in my partition, I'd add $n$ to the set $A'_{k}$ and I'd obtain a partition $\{A'_{1},A'_{2},\dots,A'_{k}\}$ in which no part contains consecutive integers.
Reciprocally, if I have a partition of $n$ into $k$ parts in which no part contains consecutive integers, I can regard the part which contains $n$ and for some element $a_i$ in it, I can pass it to the part to which belongs $a_i+1$ and afterwards, I can remove $n$, obtaining an unrestricted partition of $[n−1]$ into $k−1$ parts. How can I improve this argument?