Considering the following piece of lecture:
Construction of symmetrized wave functions.
Consider a generic quantum state of N identical particles. The exchange of two particles, represented by
$\psi= \Psi (\vec x_1,... \vec x_j,...\vec x_i,...\vec x_N ) \implies \psi'= P_{ji} \psi(\vec x_1,... \vec x_j,...\vec x_i,...\vec x_N) =\psi (\vec x_1,... \vec x_i,...\vec x_j,...\vec x_N)$
is the basic process that allows one to construct wavefunctions exhibiting a defined symmetry character. The permutation operator $P_{ji}$ effects the exchange of particles j and i by permuting the relevant positions $x_j \leftrightarrow x_i $.
The repeated action of permutation operators Pji allows one to construct both symmetric and antisymmetric wave functions. For example, with N = 3 particles
$\Phi_s(\vec x_1,\vec x_2,\vec x_3 ) = \dfrac{1}{\sqrt3!}(1+P_{13}P_{12}+P_{23}P_{12}+ P_{12}+P_{23}+P_{13}) \psi(\vec x_1,\vec x_2,\vec x_3 )\tag1$
$\Phi_s(\vec x_1,\vec x_2,\vec x_3 ) = \dfrac{1}{\sqrt3!}(\psi(\vec x_1,\vec x_2,\vec x_3 )+\psi(\vec x_2,\vec x_3,\vec x_1 )+\psi(\vec x_3,\vec x_1,\vec x_2 )+\psi(\vec x_2,\vec x_1,\vec x_3 )+\psi(\vec x_1,\vec x_3,\vec x_2 )+\psi(\vec x_3,\vec x_2,\vec x_1 ))\tag2$
My question It is not clear to me how can I come up directly with (1). While I can easily write (2) listing all the possibilities and I wouldn't miss any, I don't know how they write (1) without having written (2) first using the permutation operator directly without risk of repeating or missing a term For the specific case of 3 particles, it is not clear to me that among the $6$ terms $1,P_{13}P_{12},P_{23}P_{12}, P_{12},P_{23},P_{13}$ I am not leaving out or repeating something unless I write as (2).
All I can do is start by writting the identity and the exchanges between each pair of distinct particles:$1,P_{12},P_{23},P_{13}$ , after that I am not sure how to methodologically proceed to write the other two terms without omissions or repetitions
Is there a good strategy/algorithm to do this? The situation just worsens if I want to write (1) for a larger number of particles or if given (2) I want to write (1).