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

Abstract

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.