Chain Polynomials of Distributive Lattices are 75% Unimodal

Abstract

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 $$c_0 < \ldots < c_{\lfloor{\ell /2}\rfloor} \quad\hbox{ and }\quad c_{\lfloor{3 \ell /4}\rfloor}>\ldots>c_{\ell}.$$ This proves 75% of the inequalities implied by the Neggers unimodality conjecture.