Given a multi-variable polynomial $F$, denote the number of monomials by $N(F)$. Take for instance, \begin{align*}N(x(x+y)+(x+y)^2-(x-y)^2)=N(x^2+5xy)&=2 \qquad \text{and} \\ N((x+z)(x+y)^2)=N(x^3 + 2x^2y + x^2z + xy^2 + 2xyz + y^2z)&=6.\end{align*} Denote the symmetric group on $n$ letters $\{1,2,\dots,n\}$ by $\mathfrak{S}_n$. Define the action of $\mathfrak{S}_n$ on a function $F(x_1,\dots,x_n)$ in a natural way: given $w\in\mathfrak{S}_n$, then $w\cdot F(x_1,\dots,x_n)=F(x_{w(1)},\dots,x_{w(n)})$. Introduce the (multi-variable) rational functions $$G(\mathbf{x},\mathbf{z})=\prod_{k=1}^n\frac{x_1z_1+x_2z_2+\cdots+x_kz_k}{x_k-x_{k+1}}.$$

Assuming that the symmetric groups act only on the $x$-variables, let's compute the polynomial $$G_{\mathfrak{S}_{n+1}}=\sum_{w\in\frak{S}_{n+1}}w\cdot G.$$ I would like to ask:

QUESTION.Let $C_n=\frac1{n+1}\binom{2n}n$ be the Catalan numbers. Is there a combinatorial proof that $N(G_{\mathfrak{S}_{n+1}})=C_n$?