## Henri Mühle
and Christian Krattenthaler

# The rank enumeration of certain parabolic non-crossing partitions

### (31 pages)

**Abstract.**
We consider *m*-divisible non-crossing partitions of
{1,2,...,mn} with the property that for some *t* <= *n* no block
contains more than one of the first *t* integers. We give a closed
formula for the number of multi-chains of such non-crossing partitions
with prescribed number of blocks. Building on this result, we compute
Chapoton's *M*-triangle in this setting and conjecture a combinatorial
interpretation for the *H*-triangle. This conjecture is proved for
*m*=1.

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