Refined lattice path enumeration and combinatorial reciprocity

It is well known that the set of $m$-Dyck paths with a fixed height and amount valleys counted by Fu{\ss}-Narayana numbers. In this article, we consider start at least $t$ north steps. We give exact formulas for number such height, returns (i) valleys, (ii) $x$-coordinate divisible $m$ (iii) not $m$. The enumeration combinatorially realizes $H$-triangle appearing in recent article Krattenthaler first author (Algebr. Comb. 5, 2022) context certain parabolic noncrossing partitions. Through transformation formula due to Chapoton, an explicit associated $F$-triangle. realize polynomial means generalized Schr\"oder as flats hyperplane arrangements. Along way exhibit two new combinatorial reciprocity results.