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

Jonathan D. Farley


April 23,
refreshments at 3:45pm


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.

Speaker's Contact Info: N/A

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Combinatorics Seminar, Mathematics Department, MIT, sara(at-sign)math.mit.edu

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