Clocks are $e$-positive

Abstract

Along with his confirmation of the $e$-positivity of all cycle-chord graphs $\hat{I}{¸}_{ab1}$, the third author conjectured the $e$-positivity of all theta graphs $\hat{I}{¸}_{abc}$. In this paper, we establish the $e$-positivity of all clock graphs $\hat{I}{¸}_{ab2}$ by using the composition method. The key idea is to investigate the fibers of certain partial reversal transformation on compositions with all parts at least $2$.

13 pages, 4 figures