# A Partial Unimodality Theorem and the Stanley--Neggers Conjecture: the Linear Extensions of a Naturally Labelled Poset, Enumerated by Descents

## ABSTRACT

Let \$L\$ be a finite distributive lattice. (One could think of \$L\$ as the family of order ideals---or downward-closed subsets---of a finite partially ordered set, ordered by set-inclusion.) Let \$f_i\$ be the number of \$i+1\$-element chains (totally ordered subsets) of \$L\$.

One of the main implications of the Stanley--Neggers Conjecture is that the \$f\$-vector---the 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 ``three-quarters 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 bounded-homomorphic image of a finitely-generated free lattice, a so-called ``bounded" lattice in the sense of McKenzie.

We present some vague ideas for constructing a counter-example to the Stanley--Neggers conjecture.

Polytopes, simplicial complexes, and matchings arise.

The talk contains interesting open problems for graduate students and will be accessible to a general audience.

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