Chain and antichain enumeration in posets, and b-ary partitions
Name
56019029-MIT.pdf
Description
Full printable version
Size
2.11 MB
Format
Adobe PDF
Checksum (MD5)
207e949db8ac822972c5227e7971bb34
Author(s)
Early, Edward Fielding, 1977-
Advisor(s)
Richard P. Stanley.
Date Issued
2004
Publisher
Massachusetts Institute of Technology
Abstract
The Greene-Kleitman theorem says that the lengths of chains and antichains in any poset are intimately related via an integer partition, but very little is known about the partition [lambda](P) for most posets P. Our first goal is to develop a method for calculating values of [lambda]k(P) for certain posets. We find the size of the largest union of two or three chains in the lattice of partitions of n under dominance order, and in the Tamari lattice. Similar techniques are then applied to the k-equal partition lattice. We also present some partial results and conjectures on chains and antichains in these lattices. We give an elementary proof of the rank-unimodality of L(2, n, m), and find a symmetric chain decomposition of L(2, 2, m). We also present some partial results and conjectures about related posets, including a theorem on the size of the largest union of k chains in these posets and a bijective proof of the symmetry of the H-vector for 2 x n. We answer a question of Knuth about the existence of a Gray path for binary partitions, and generalize to b-ary partitions when b is even. We also discuss structural properties of the posets Rb(n), and compute some chain and antichain lengths in the subposet of join-irreducibles.
Description
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004.
Includes bibliographical references (leaves 69-72).
Subjects
Mathematics.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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