Chain and antichain enumeration in posets, and b-ary partitions
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004.
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| Format: | Thesis |
| Language: | eng |
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Massachusetts Institute of Technology
2006
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| Online Access: | http://hdl.handle.net/1721.1/30148 |
| _version_ | 1811089744327606272 |
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| author | Early, Edward Fielding, 1977- |
| author2 | Richard P. Stanley. |
| author_facet | Richard P. Stanley. Early, Edward Fielding, 1977- |
| author_sort | Early, Edward Fielding, 1977- |
| collection | MIT |
| description | Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004. |
| first_indexed | 2024-09-23T14:24:01Z |
| format | Thesis |
| id | mit-1721.1/30148 |
| institution | Massachusetts Institute of Technology |
| language | eng |
| last_indexed | 2024-09-23T14:24:01Z |
| publishDate | 2006 |
| publisher | Massachusetts Institute of Technology |
| record_format | dspace |
| spelling | mit-1721.1/301482019-04-12T09:37:34Z Chain and antichain enumeration in posets, and b-ary partitions Early, Edward Fielding, 1977- Richard P. Stanley. Massachusetts Institute of Technology. Dept. of Mathematics. Massachusetts Institute of Technology. Dept. of Mathematics. Mathematics. 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. 2006-03-24T18:23:59Z 2006-03-24T18:23:59Z 2004 2004 Thesis http://hdl.handle.net/1721.1/30148 56019029 eng M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission. http://dspace.mit.edu/handle/1721.1/7582 72 leaves 2213961 bytes 2213767 bytes application/pdf application/pdf application/pdf Massachusetts Institute of Technology |
| spellingShingle | Mathematics. Early, Edward Fielding, 1977- Chain and antichain enumeration in posets, and b-ary partitions |
| title | Chain and antichain enumeration in posets, and b-ary partitions |
| title_full | Chain and antichain enumeration in posets, and b-ary partitions |
| title_fullStr | Chain and antichain enumeration in posets, and b-ary partitions |
| title_full_unstemmed | Chain and antichain enumeration in posets, and b-ary partitions |
| title_short | Chain and antichain enumeration in posets, and b-ary partitions |
| title_sort | chain and antichain enumeration in posets and b ary partitions |
| topic | Mathematics. |
| url | http://hdl.handle.net/1721.1/30148 |
| work_keys_str_mv | AT earlyedwardfielding1977 chainandantichainenumerationinposetsandbarypartitions |