2026-09-20T22:20:59Zhttps://dspace.mit.edu/server/oai/requestoai:dspace.mit.edu:1721.1/1046032026-06-17T14:42:42Zcom_1721.1_7582com_1721.1_7581col_1721.1_131022
Richard P. Stanley.
Engel Shaposhnik, Efrat
Massachusetts Institute of Technology. Department of Mathematics.
Massachusetts Institute of Technology. Department of Mathematics
2016-09-30T19:37:53Z
2016-09-30T19:37:53Z
2016
2016
http://hdl.handle.net/1721.1/104603
958839493
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2016.
Cataloged from PDF version of thesis.
Includes bibliographical references (page 87).
This thesis consists of two parts. In the first part we count antichains of interval orders and in particular semiorders. We associate a Dyck path to each interval order, and give a formula for the number of antichains of an interval order in terms of the corresponding Dyck path. We then use this formula to give a generating function for the total number of antichains of semiorders, enumerated by the sizes of the semiorders and the antichains. In the second part we expand the work of Liu and Stanley on Dilworth lattices. Let L be a distributive lattice, let -(L) be the maximum number of elements covered by a single element in L, and let K(L) be the subposet of L consisting of the elements that cover o-(L) elements. By a result of Dilworth, K(L) is also a distributive lattice. We compute o(L) and K(L) for various lattices L that arise as the coordinate-wise partial ordering on certain sets of semistandard Young tableaux.
by Efrat Engel Shaposhnik.
Ph.D.
87 pages
eng
Massachusetts Institute of Technology
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
Mathematics.
Antichains of interval orders and semiorders, and Dilworth lattices of maximum size antichains
Thesis
application/pdf
false
Publication
unknown
unknown
<Publication xmlns="https://www.openaire.eu/cerif-profile/1.1/" id="6c448406-c7b9-4395-aebc-74254cf1a69e">
<Type xmlns="https://www.openaire.eu/cerif-profile/vocab/COAR_Publication_Types">http://purl.org/coar/resource_type/c_1843</Type>
<Language>eng</Language>
<Title>Antichains of interval orders and semiorders, and Dilworth lattices of maximum size antichains</Title>
<PublishedIn>
<Publication>
</Publication>
</PublishedIn>
<PublicationDate>2016</PublicationDate>
<Authors>
<Author>
<DisplayName>Engel Shaposhnik, Efrat</DisplayName>
<Affiliation>
<OrgUnit>
</OrgUnit>
</Affiliation>
</Author>
</Authors>
<Editors>
</Editors>
<Publishers>
<Publisher>
<DisplayName>Massachusetts Institute of Technology</DisplayName>
<OrgUnit />
</Publisher>
</Publishers>
<License>http://dspace.mit.edu/handle/1721.1/7582</License>
<Keyword>Mathematics.</Keyword>
<Abstract>This thesis consists of two parts. In the first part we count antichains of interval orders and in particular semiorders. We associate a Dyck path to each interval order, and give a formula for the number of antichains of an interval order in terms of the corresponding Dyck path. We then use this formula to give a generating function for the total number of antichains of semiorders, enumerated by the sizes of the semiorders and the antichains. In the second part we expand the work of Liu and Stanley on Dilworth lattices. Let L be a distributive lattice, let -(L) be the maximum number of elements covered by a single element in L, and let K(L) be the subposet of L consisting of the elements that cover o-(L) elements. By a result of Dilworth, K(L) is also a distributive lattice. We compute o(L) and K(L) for various lattices L that arise as the coordinate-wise partial ordering on certain sets of semistandard Young tableaux.</Abstract>
<Access xmlns="http://purl.org/coar/access_right"
>
</Access>
</Publication>