We give a simple and natural proof of (an extension of) the identity P(k, l, n)=P_2(k-1, l-1, n-1). The number P(k, l, n) counts noncrossing partitions of {1, 2, ..., l} into n parts such that no part contains two numbers x and y, 0<y-x<k. The lower index 2 indicates partitions with no part of size three or more. We use the identity to give quick proofs of the closed formulae for P(k, l, n) when k is 1, 2, or 3.

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