Diagrammatic Okada monoid and cellularity of the Okada algebra

It is well known that the Young lattice is the Bratelli diagram of the symmetric groups, expressing how irreducible representations restrict from $\mathfrak{S}_{N}$ to $\mathfrak{S}_{N-1}$. In 1975, Stanley discovered a similar lattice called the Young-Fibonacci lattice which was identified as the B