Bijection between $n$-partitions and "flattened" canonical $n+1$-partitions The set of $n$-partitions (partitions of the set $\{1, 2, \ldots, n\}$) and the set of "flattened" canonical $(n+1)$-partitions (those permutations obtained by removing the bars from an $(n+1)$ partition written in canonical form) are both counted by the Bell number $B(n)$. Can you give an intuitive bijection between these two sets?
 A: In one of our recent article (see link below Section 4):
https://www.researchgate.net/publication/344683169_Run_Distribution_Over_Flattened_Partitions
we gave a bijection between set partitions of $[n]$ and flattened partitions of length $[n+1]$.
For example: Let us consider a partition $P = 12|3|45$ of $[5]$. Then re-ordering the entries of $P$ such that in each block, the smallest entry appears at the end, then we would have a new partition $P' = 21|3|54$. Increasing by 1, each of the entries in $P'$ gives $P* = 32|4|65$. The corresponding flattened partition is obtained by appending a "1" before $P*$ and ignoring the vertical marks "|" separating the blocks, i.e., $1P* = 132465$, which is a flattened partition of length 6.
Moreover, using now the flattened partition $1P* = 132465$, we can get back $P$ by inserting the vertical mark "|" immediately after each right to left minimum gives $1|32|4|65|$. Deleting 1 and re-ordering the remaining
gives the partition $P = 12|3|45$.
For technicalities and details of the bijection, please refer to the above link.
