  A natural unit interval order is a naturally labelled partially ordered set
that avoids patterns ${\bf 3} + {\bf 1}$ and $\bf 2 + 2$. To each natural unit
interval order one can associate a symmetric function. The Stanley-Stembridge
conjecture states that each such symmetric function is positive in the basis of
complete homogenous symmetric functions. This conjecture has connections to
cohomology rings of Hessenberg varieties, and to Kazhdan-Lusztig theory. We use
a diagrammatic technique to re-prove the special case of the conjecture for
unit interval orders additionally avoiding pattern $\bf 2 + 1 + 1$. Originally
this special case is due to Gebhard and Sagan.
