|P(d)(n,k)| = |P(d-j)(n-j,k-j)|,
where P(d)(n,k)
is the collection of all set partitions of
[n]:={1,2,...,n} into k blocks such that for any two
distinct elements x,y in the same block, we have
|y-x| >= d. We
also generalize an identity of Klazar on d-regular noncrossing
partitions. Namely, we show that the number of d-regular
l-noncrossing partitions of [n] is equal to the number of
(d-1)-regular enhanced l-noncrossing partitions of
[n-1].
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