Analogues of the elementary symmetric polynomials for the alternating group In the case of three variables, the elementary symmetric polynomials are
$$ \begin{align}
e_1(X_1,X_2,X_3)&:=X_1+X_2+X_3, \\
e_2(X_1,X_2,X_3)&:=X_1 X_2+X_1 X_3+X_2 X_3, \\ 
e_3(X_1,X_2,X_3)&:=X_1 X_2 X_3. \\
\end{align}$$
Knowledge of the values of $e_1,e_2,e_3$ determines the variables $X_1,X_2,X_3$ up to any permutation of $S_3$. That is, if 
$$
\begin{align} e_1(X_1,X_2,X_3)&=e_1(Y_1,Y_2,Y_3), \\ e_2(X_1,X_2,X_3)&=e_2(Y_1,Y_2,Y_3) ,\\
e_3(X_1,X_2,X_3)&=e_2(Y_1,Y_2,Y_3),\\  \end{align}$$
then there exists a permutation $\sigma \in S_3$ such that $$X_i=Y_\sigma(i) ,$$
for all $1 \leq i \leq 3$.
I'm curious as to whether there exist other "less symmetric" polynomials, say $\{P_n(X_1,X_2,X_3)\}_n$ such that having
$$P_n(X_1,X_2,X_3)=P_n(Y_1,Y_2,Y_3) $$for all $n$ implies that there exists an even permutation $\sigma \in A_3 \subsetneq S_3$ for which $X_i=Y_\sigma(i)$ for all $1 \leq i \leq 3$.
I have tried keeping two of the elementary symmetric polynomials, replacing the third, but that didn't work out. 
I would appreciate help with finding such polynomials $P_n$ (if they exist). Thank you! 
 A: You're almost there. It suffices to add a fourth polynomial :
$$
\begin{array}{lcl}
P_1 &=& X_1+X_2+X_3 \\
P_2 &=& X_1X_2+X_1X_3+X_2X_3 \\
P_3 &=& X_1X_2X_3 \\
P_4 &=& X_1^2X_2+X_2^2X_3+X_3^2X_1
\end{array}
$$
Suppose that $X=(X_1,X_2,X_3)$ and $Y=(Y_1,Y_2,Y_3)$ satisfy $P_n(X)=P_n(Y)$ for $1\leq n \leq 4$. Using the first three equalities only, we already know that there is a $\sigma\in {\mathfrak S}_n$ such that $X=Y_\sigma$ (by which I mean that $X_i=Y_{\sigma(i)}$ for $1\leq i \leq 3$).
If $\sigma$ is already even, we are done. Otherwise, $\sigma$ is a transposition, say 
$\sigma=(1,2)$. Then the last identity becomes $P_4(Y_2,Y_1,Y_3)=P_4(Y_1,Y_2,Y_3)$. Now the polynomial $D=P_4(Y_2,Y_1,Y_3)-P_4(Y_1,Y_2,Y_3)$ factorizes as
$$
D=(Y_2-Y_1)(Y_3-Y_1)(Y_3-Y_2)
$$
So at least two $Y_i$'s are equal. It is easy to deduce from here that there are other permutations $\gamma$ satisfying $X=Y_\gamma$, some of which are even.
A: Here’s one way to do it: to get a list of polynomials for any $n$, start with the list of elementary symmetric polynomials on $n$ variables and add in the Vandermonde polynomial on $n$ variables.
