A simple bijection for enhanced,classical, and 2-distant k-noncrossing partitions

In this note, we give a simple extension map from partitions of subsets of $n]$ to partitions of $n+1]$, which sends $\delta$-distant $k$-crossings to $(\delta +1)$-distant $k$-crossings (and similarly for nestings). This map provides a combinatorial proof of the fact that the numbers of enhanced, classical, and 2-distant $k$-noncrossing partitions are each related to the next via the binomial transform. Our work resolves a recent conjecture of Zhicong Lin and generalizes earlier reduction identities for partitions.