The rank enumeration of certain parabolic non-crossing partitions

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 integers 1, 2, ... , t . 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.