Given $d\in \mathbb{N}$, let $X_d:= \{(\ell_1, \ldots , \ell_d): 0 \le \ell_1 \le \ldots \le \ell_d \le d\}\subset \mathbb{Z}^d$, and endow $X_d$ with the (usual) partial order, namely, $x\le y$ if and only if $x_j \le y_j$ for all $j=1,\ldots , d$. Note that if $Y\subset X_d$ is linearly ordered then $\# Y \le d^2 + 1$. Let $N_d$ be the number of linearly ordered subsets of $X_d$ of maximal length, that is, $N_d :=\# \{Y\subset X_d : Y \mbox{ linearly ordered, } \# Y = d^2 + 1\}$.
My question: Is there any neat expression available for $N_d$? If not, perhaps at least the asymptotics of $N_d$ as $d\to \infty$ can be worked out? Any answers or pertinent references will be much appreciated.