the advantage that one can derive recursive formulas for $\mathcal{V}_{s,n}^{m'\rightarrow m}(q)$ (see for example [@CSVW14 Theorem 2.2]) but closed formulas were established only in the case of $m'=1,m=2$ [@CSSW14 Theorem 3.3] and $m'=1,m=3$ [@CSVW14 Section 1.6]. Our approach uses a different short exact sequence (see Section \[section6\]) which enables us to give a combinatorial formula in the most general setting.

Combinatorics of Dyck paths {#section3}
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In this section we introduce our combinatorial model and certain combinatorial statistics which will be needed to describe the polynomials .

 {#section31}

A Dyck path is a diagonal lattice path from the origin $(0,0)$ to $(s,n)$ for some non–negative integrs $s,n\in {\mathbb Z}_+$, such that the path never goes below the x–axis. We encode such a path by a 01–word, where $1$ encodes the up–steps and $0$ the down–steps. For a path $P$ we set $$\text{supp}(P):=\{z\in {\mathbb Z}_+^2 : z \text{ is a point on $P$}\}.$$ The length of a point $z$ on a path $P$ is defined to be its $x$–coordinate and its $y$–coordinate is called the height; we will frequently use the notation $z=(\ell_z(P),\text{ht}_z(P))$. We denote by $\mathcal{D}_{n}$ the set of Dyck paths that end at height $n$ and by $\mathcal{D}_n(s)$ the 