Chain polynomials of distributive lattices are 75 % unimodal
  It is shown that the numbers $c_i$ of chains of length $i$ in the proper part $L\setminus\{0,1\}$ of a distributive lattice $L$ of length $\ell +2$ satisfy the inequalities $$c_0<...<c_{\lfloor{\ell /2}\rfloor} \quad{and}\quad c_{\lfloor{3 \ell /4}\rfloor}>...>c_{\ell}.$$ This proves 75 % of the inequalities implied by the Neggers unimodality conjecture. 
Anders Bj\"orner and Jonathan David Farley