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

6 pages