  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.
