Garside combinatorics for Thompson’s monoid $F^+$ and a hybrid with the braid monoid $B_{\infty }^{+}$

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On the model of simple braids, defined to be the left divisors of Garside's elements $\Delta_n$ in the monoid $B_\infty^+$ , we investigate simple elements in Thompson's monoid $F^+$ and in a larger monoid $H^+$ that is a hybrid of $B_\infty^+$ 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.