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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 mapprovides a combinatorial proof of the fact that the numbers of enhanced,classical, and 2-distant k-noncrossing partitions are each related to the nextvia the binomial transform. Our work resolves a recent conjecture of ZhicongLin and generalizes earlier reduction identities for partitions.

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