Building a many-body basis

There are \( \left ( \begin{array}{c} 4 \\ 2 \end{array} \right) = 6 \) two-particle states, which we list with the total \( M \):

Occupied \( M \)
1,2 0
1,3 -1
1,4 0
2,3 0
2,4 1
3,4 0
There are 4 states with \( M= 0 \), and 1 each with \( M = \pm 1 \).