  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 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 $m=1$.
