defined by the recurrence relation $$U_n(x)=2x U_{n-1}(x)-U_{n-2}(x),\ \ n\geq 2,$$ with initial data $U_0(x)=1$ and $U_1(x)=2x$. Define polynomials $$p_n(x):=x^{n/2}U_n((2\sqrt{x})^{-1})=\sum_{s=0}^{\lfloor\frac{k}{2}\rfloor} (-1)^s {\genfrac[]{0pt}0{k-s}{s}}.$$ The first equality of the next lemma follows from [@GX05 Lemma 4.2] and the second equality has been proved in [@BCSV15 Corollary 1.3].

\[Ges\] We have $\mathcal{G}^m_n(x)=0$ if $m<n$ and otherwise $$\mathcal{G}^m_n(x)=\frac{p_{m-n}(x)}{p_{m+1}(x)}=\mathcal{A}_{n}^{1\rightarrow m+1}(x,1).$$ In particular, $$\mathcal{V}_{s,n}^{1\rightarrow m+1}(1)=|\mathcal{D}^m_n(s)|,\ \ \forall n\leq m.$$

This result suggests a deeper connection between graded multiplicities in Demazure flags and certain Dyck path statistics and the motivation of the present paper is to determine this connection. As a byproduct we will generalize the above lemma and find a combinatorial model (in terms of Dyck paths) whose generating series is $\mathcal{A}^{1\rightarrow m}_n(x,1)$ (even for $n\geq m$) (see Proposition \[numerical\]).

 {#admissibledef}

A point $(z_1,z_2)\in \text{supp}(P)$ is called a peak (resp. valley) of the path if $(z_1\pm 1,z_2-1)\in \text{supp}(P)$ (resp. $(z_1\pm 1,z_2+1)\in \text{supp}(P)$). By convention, we will also call the endpoint of a path which is immediately preceded by a down–step a valley. For example, the path in Figure \[fig1\] has three peaks and three valleys. Given a pair of non-negative integers $(a,b)\in {\mathbb Z}^2_+$, we say that $P\in \mathcal{D}_n$ is $(a,b)$–admissible if and only if $P$ satisfies 