I was looking at the proof of Selberg's Integral Formula, which is given below:
Selberg Integral Formula
Let $$\Delta(x_1,\ \cdots,\ x_n)\equiv\Delta(\vec{x}) = \prod_{1\le i<j\le n}(x_j-x_i)$$ and $$\phi_n^{(\alpha,\ \beta,\ \gamma)}(x_1,\ \cdots,\ x_n)\equiv\phi(\vec{x}) = \Delta(\vec{x})^{2\gamma}\prod_{i=1}^nx_i^{\alpha-1}(1-x_i)^{\beta-1}$$ Then $$ S_n(\alpha,\ \beta,\ \gamma) = \int_{[0,\ 1]^n}\phi_n^{(\alpha,\ \beta,\ \gamma)}(\vec{x})\text{d}\vec{x} = \prod_{j=0}^{n-1}\frac{\Gamma(1+(1+j)\gamma)\Gamma(\alpha+j\gamma)\Gamma(\beta+j\gamma)}{\Gamma(1+\gamma)\Gamma(\alpha+\beta+(n+j-1)\gamma)} $$
The proof I was reading is from Mehta's book. One of the first steps in the proof is to assume that $\gamma$ is an integer and then write $$ \Delta(\vec x)^{2\gamma} = \sum_{\substack{j_1\le j_2\le\cdots\le j_n \\ j_1+j_2+\cdots+j_n=2\gamma{n\choose 2}}}c(j_1,\ j_2,\ \cdots,\ j_n)x^{j_1}x^{j_2}\cdots x^{j_n} $$ How exactly is it justified to order the $j_i$? To give an example of why this is so concerning, consider taking $n = 2$, $\gamma = 1$, then $$ |\Delta(x)|^{2\gamma}=(x_2-x_1)^2=x_2^2-2x_1x_2+x_1^2 $$ the 2-term partitions of $2\gamma{n\choose2}$ are $0,\ 2$, $1,\ 1$, and $2,\ 0$. These correspond to the $x_2^2$ term, the $-2x_1x_2$ term, and the $x_1^2$ term respectively. If we restrict the partitions to those that are strictly ordered, like Selberg is attempting to due, we omit the $x_1^2$ term, but $$ |\Delta(x)|^{2\gamma} \neq x_2^2-2x_1x_2 $$ So how could this even be correct?
I don't think what you wrote is quite correct, and it's not what is written here on page 2 (or 490 according to the BAMS numbering). What you want to do is expand even powers of the Vandermonde determinant in terms of a basis of monomials. Hence Selberg wrote$$\sum_{k_1, \ldots, k_n} c_{k_1, \ldots, k_n} t_1^{k_1} \ldots t_n^{k_n}.$$Of course, since the Vandermonde determinant is a skew-symmetric function, the even powers are symmetric functions, so you can certainly restrict your sum to an ordered sum but then you need to replace ordinary monomials by monomial symmetric functions (which form the simplest of all bases of the ring of symmetric functions).