Garside combinatorics for Thompson's monoid $F^{+}$ and a hybrid with the braid monoid $B\mathrm{\_}{\mathrm{\infty }}^{+}$

On the model of simple braids, defined to be the left divisors of Garside’s elements $\mathrm{\Delta }\mathrm{\_}n$ in the monoid $B\mathrm{\_}{\mathrm{\infty }}^{+}$ , we investigate simple elements in Thompson’s monoid $F^{+}$ and in a larger monoid $H^{+}$ that is a hybrid of $B\mathrm{\_}{\mathrm{\infty }}^{+}$ and $F^{+}$ : in both cases, we count how many simple elements left divide the right lcm of the first n – 1 atoms, and characterize their normal forms in terms of forbidden factors. In the case of $H^{+}$, a generalized Pascal triangle appears.