A Partial Unimodality Theorem and the StanleyNeggers Conjecture: the Linear Extensions of a Naturally Labelled Poset, Enumerated by DescentsJonathan D. FarleyMIT
April 23,

ABSTRACT


Let $L$ be a finite distributive lattice. (One could think of $L$ as the family of order idealsor downwardclosed subsetsof a finite partially ordered set, ordered by setinclusion.) Let $f_i$ be the number of $i+1$element chains (totally ordered subsets) of $L$. One of the main implications of the StanleyNeggers Conjecture is that the $f$vectorthe sequence $f_{1}$, $f_0$, $f_1$, \dots, $f_h$is unimodal, where $h$ is the ``rank" or height of the lattice $L$. We prove that the unimodality conjecture is ``threequarters true" (joint work with Anders Bj\"orner): that is, the first half of the sequence is increasing, and the last quarter of the sequence is decreasing. We use the fact that the symmetric group $S_h$ with the weak Bruhat order is a boundedhomomorphic image of a finitelygenerated free lattice, a socalled ``bounded" lattice in the sense of McKenzie. We present some vague ideas for constructing a counterexample to the StanleyNeggers conjecture. Polytopes, simplicial complexes, and matchings arise. The talk contains interesting open problems for graduate students and will be accessible to a general audience. 
Combinatorics Seminar, Mathematics Department, MIT, sara(atsign)math.mit.edu 

