Building a many-body basis

There are \( \left ( \begin{array}{c} 6 \\ 2 \end{array} \right) = 15 \) two-particle states, which we list with the total \( M \):

Occupied \( M \) Occupied \( M \) Occupied \( M \)
1,2 -4 2,3 -2 3,5 1
1,3 -3 2,4 -1 3,6 2
1,4 -2 2,5 0 4,5 2
1,5 -1 2,6 1 4,6 3
1,6 0 3,4 0 5,6 4

There are 3 states with \( M= 0 \), 2 with \( M = 1 \), and so on.