Adjoint strings

This is a draft!! Just wrote it up to record some thoughts until I can find time to fill in the explanations for what is going on. \[\begin{array}{c} &&&& \mathbf 3 \\ (01) & \dashv & (011) & \dashv & (02) & \dashv & (001) & \dashv & (12) \\ &&&& \mathbf 2 \\ && (0) & \dashv & (00) & \dashv & (1)    \\ &&&& \mathbf 1    \\    &&&&      \\    &&&&  \mathbf 0    \\     \end{array}\]