Another Family of Permutations Counted by the Bell Numbers

Using a permutation code introduced by Rakotondrajao and the second author, we associate with every set partition of $[n]$ a permutation over $[n]$, thus defining a class of permutation whose size is the $n$-th Bell number. We characterize the permutations belonging to this class and we study the distribution of weak exceedances over these permutations, which turns out to be enumerated by the Stirling numbers of the second kind. We provide a direct bijection between our class of permutations and another equisized class of permutations introduced by Poneti and Vajnovszki.