On a rank-unimodality conjecture of Morier-Genoud and Ovsienko

Let ?=(a,b,…) be a composition. Consider the associated poset F(?), called fence, whose covering relations arex1?x2?…?xa+1?xa+2?…?xa+b+1?xa+b+2?…. We study distributive lattice L(?) consisting of all lower order ideals F(?). These lattices are important in theory cluster algebras and their rank generating functions can used to define q-analogues rational numbers. In particular, we make progress on recent conjecture Morier-Genoud Ovsienko that is unimodal. show if one parts ? greater than sum others, then true. enjoys stronger properties having nested chain decomposition sequence which either top or bottom interlacing, latter being recently defined property sequences. verify these hold for compositions with at most three what call d-divided posets, generalizing work Claussen simplifying construction Gansner.