On the rank function of a differential poset
We study r-differential posets, a class of combinatorial objects introduced in 1988 by the first author, which gathers together a number of remarkable combinatorial and algebraic properties, and generalizes important examples of ranked posets, including the Young lattice. We first provide a simple b...
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| Other Authors: | |
| Format: | Article |
| Language: | en_US |
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International Press
2012
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| Online Access: | http://hdl.handle.net/1721.1/71661 https://orcid.org/0000-0003-3123-8241 |
| _version_ | 1811078068408680448 |
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| author | Stanley, Richard P. Zanello, Fabrizio |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Stanley, Richard P. Zanello, Fabrizio |
| author_sort | Stanley, Richard P. |
| collection | MIT |
| description | We study r-differential posets, a class of combinatorial objects introduced in 1988 by the first author, which gathers together a number of remarkable combinatorial and algebraic properties, and generalizes important examples of ranked posets, including the Young lattice. We first provide a simple bijection relating differential posets to a certain class of hypergraphs, including all finite projective planes, which are shown to be naturally embedded in the initial ranks of some differential poset. As a byproduct, we prove the existence, if and only if r≥6, of r-differential posets nonisomorphic in any two consecutive ranks but having the same rank function. We also show that the Interval Property, conjectured by the second author and collaborators for several sequences of interest in combinatorics and combinatorial algebra, in general fails for differential posets. In the second part, we prove that the rank function p[subscript n] of any arbitrary r-differential poset has nonpolynomial growth; namely, ... a bound very close to the Hardy-Ramanujan asymptotic formula that holds in the special case of Young's lattice. We conclude by posing several open questions. |
| first_indexed | 2024-09-23T10:52:52Z |
| format | Article |
| id | mit-1721.1/71661 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T10:52:52Z |
| publishDate | 2012 |
| publisher | International Press |
| record_format | dspace |
| spelling | mit-1721.1/716612022-09-27T15:38:26Z On the rank function of a differential poset Stanley, Richard P. Zanello, Fabrizio Massachusetts Institute of Technology. Department of Mathematics Stanley, Richard P. Stanley, Richard P. We study r-differential posets, a class of combinatorial objects introduced in 1988 by the first author, which gathers together a number of remarkable combinatorial and algebraic properties, and generalizes important examples of ranked posets, including the Young lattice. We first provide a simple bijection relating differential posets to a certain class of hypergraphs, including all finite projective planes, which are shown to be naturally embedded in the initial ranks of some differential poset. As a byproduct, we prove the existence, if and only if r≥6, of r-differential posets nonisomorphic in any two consecutive ranks but having the same rank function. We also show that the Interval Property, conjectured by the second author and collaborators for several sequences of interest in combinatorics and combinatorial algebra, in general fails for differential posets. In the second part, we prove that the rank function p[subscript n] of any arbitrary r-differential poset has nonpolynomial growth; namely, ... a bound very close to the Hardy-Ramanujan asymptotic formula that holds in the special case of Young's lattice. We conclude by posing several open questions. Massachusetts Institute of Technology. Dept. of Mathematics 2012-07-17T19:03:28Z 2012-07-17T19:03:28Z 2012-04 2011-11 Article http://purl.org/eprint/type/JournalArticle 1097-1440 1077-8926 http://hdl.handle.net/1721.1/71661 Stanley, Richard P. and Fabrizio Zanello. "On the rank function of a differential poset." Electronic Journal of Combinatorics (2012) 19.2, p.1-17. https://orcid.org/0000-0003-3123-8241 en_US http://www.combinatorics.org/ojs/index.php/eljc/article/view/v19i2p13 Electronic Journal of Combinatorics Creative Commons Attribution-Noncommercial-Share Alike 3.0 http://creativecommons.org/licenses/by-nc-sa/3.0/ application/pdf International Press arXiv |
| spellingShingle | Stanley, Richard P. Zanello, Fabrizio On the rank function of a differential poset |
| title | On the rank function of a differential poset |
| title_full | On the rank function of a differential poset |
| title_fullStr | On the rank function of a differential poset |
| title_full_unstemmed | On the rank function of a differential poset |
| title_short | On the rank function of a differential poset |
| title_sort | on the rank function of a differential poset |
| url | http://hdl.handle.net/1721.1/71661 https://orcid.org/0000-0003-3123-8241 |
| work_keys_str_mv | AT stanleyrichardp ontherankfunctionofadifferentialposet AT zanellofabrizio ontherankfunctionofadifferentialposet |