Hawkes's degreewise maximal Dyck path formula

Let $n$ and $\ell+1$ be relatively prime positive integers. Let $\mathbf{D}_{n/(\ell+1)}$ be the set of rational Dyck paths represented by sequences $x=(x_0,\ldots,x_\ell)$ satisfying

$$x_0+\cdots+x_i<\frac{n(i+1)}{\ell+1}\quad(0\leq i\leq\ell-1),\qquad x_0+\cdots+x_\ell=n.$$

Define

$$M=\sum_{i=0}^{\ell-1}\left\lfloor\frac{n(i+1)}{\ell+1}\right\rfloor,$$ $$area(x)=M-\bigl(x_0\ell+x_1(\ell-1)+\cdots+x_{\ell-1}+x_\ell\cdot0\bigr).$$

For $0\leq i\leq j<\ell$, define $\beta_{ij}(x)=(x_i+\cdots+x_j)-\frac{n}{\ell+1}(j-i+1)$ and

$$\gamma_{ij}(x)=\begin{cases}\min\left(x_{i-1},\left\lfloor|\beta_{ij}(x)|\right\rfloor\right)&\text{if }\beta_{ij}(x)<0,\min\left(x_i,\left\lfloor|\beta_{ij}(x)|\right\rfloor\right)&\text{if }\beta_{ij}(x)>0,\end{cases}$$

with $degr(x)=\sum_{1\leq i\leq j<\ell}\gamma_{ij}(x)$. Set

$$\mathbf{D}_{n/(\ell+1)}^d=\{x\in\mathbf{D}_{n/(\ell+1)}:degr(x)=d\},$$ $$under(x)=\min_{0\leq i<\ell}\left(\frac{n(i+1)}{\ell+1}-(x_0+\cdots+x_i)\right),$$ $$\mathbf{T}_{n/(\ell+1)}^d=\{x\in\mathbf{D}_{n/(\ell+1)}:under(x)=1/(\ell+1),\ degr(x)=d\}.$$

Define

$$\mathcal{C}_{n/(\ell+1)}^d=\sum_{x\in\mathbf{D}_{n/(\ell+1)}^d}q^{area(x)}t^{M-d-area(x)},\qquad sym(a,b)=\frac{q^{b+1}t^a-q^at^{b+1}}{q-t}.$$

Hawkes's degreewise conjecture. For every $d$,

$$\mathcal{C}_{n/(\ell+1)}^d=\sum_{x\in\mathbf{T}_{n/(\ell+1)}^d}sym\bigl(area(x),M-d-area(x)\bigr).$$

This is the self-contained degreewise form of the maximal-path formula. The paper proves it in specified cases and supplies evidence, while the general statement remains open.