The Stanley-Stembridge Conjecture for $\bf 2 + 1 +1$-avoiding unit interval orders: a diagrammatic proof

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.