|*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].

Received: March 7, 2009. Accepted: May 3, 2009. Final Version: July 2, 2009.

The following versions are available:

- PDF (196 K)
- PostScript (446 K)
- DVI version
- Tex version