2026-09-19T00:49:09Zhttps://dspace.mit.edu/server/oai/requestoai:dspace.mit.edu:1721.1/301482022-01-13T07:54:35Zcom_1721.1_7582com_1721.1_7581col_1721.1_131022
Richard P. Stanley.
Early, Edward Fielding, 1977-
Massachusetts Institute of Technology. Dept. of Mathematics.
Massachusetts Institute of Technology. Department of Mathematics
2006-03-24T18:23:59Z
2006-03-24T18:23:59Z
2004
2004
http://hdl.handle.net/1721.1/30148
56019029
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004.
Includes bibliographical references (leaves 69-72).
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.
by Edward Fielding Early.
Ph.D.
72 leaves
2213961 bytes
2213767 bytes
application/pdf
application/pdf
eng
Massachusetts Institute of Technology
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http://dspace.mit.edu/handle/1721.1/7582
Mathematics.
Chain and antichain enumeration in posets, and b-ary partitions
Thesis
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Publication
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