Catalan sequence problem

Question

• For $4$ variables $g_{1}, g_{2}, g_{3}, g_{4}$ ,the sum of the expressions $$ (g_1-g_2)(g_3-g_4),(g_1-g_3)(g_4-g_2),(g_1-g_4)(g_2-g_3)\quad \mbox{is}\quad {\large 0} $$ ( known as Euler's Identity). So the rank of the vector space is $2$.
• For $6$ variables, let \begin{align} & M = \left\{(g_a-g_b)(g_c-g_d)(g_e-g_f)\mid a,b,c,d,e,f\text{ is a} \right. \\[1mm] & \left.\text{permutation of 1,2,3,4,5,6, and }a < b,c < d,e < f\right\} \end{align} It is easy to check by hand that the rank of $M$ is $5$.
• For $8,10$ variables, the rank is $14, 42$ respectively. ( I found it using $\tt Mathematica$).
• For $2n$ variables, I guess the rank is the $\left(n + 1\right)$th Catalan Number. Is that right?