The rank enumeration of certain parabolic non-crossing partitions

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