Unimodality of partitions with distinct parts inside Ferrers shapes
We investigate the rank-generating function F[subscript λ] of the poset of partitions contained inside a given shifted Ferrers shape λ. When λ has four parts, we show that F [subscript λ] is unimodal when λ=〈n, n-1, n-2, n-3〉, for any n≥4, and that unimodality fails for the doubly-indexed, infinite...
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| Format: | Article |
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Elsevier
2017
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| Online Access: | http://hdl.handle.net/1721.1/112204 https://orcid.org/0000-0003-3123-8241 |
| _version_ | 1811093083180236800 |
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| author | Zanello, Fabrizio Stanley, Richard P |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Zanello, Fabrizio Stanley, Richard P |
| author_sort | Zanello, Fabrizio |
| collection | MIT |
| description | We investigate the rank-generating function F[subscript λ] of the poset of partitions contained inside a given shifted Ferrers shape λ. When λ has four parts, we show that F [subscript λ] is unimodal when λ=〈n, n-1, n-2, n-3〉, for any n≥4, and that unimodality fails for the doubly-indexed, infinite family of partitions of the form λ=〈n, n-t, n-2t, n-3t〉, for any given t≥2 and n large enough with respect to t.When λ has b≤3 parts, we show that our rank-generating functions F[subscript λ] are all unimodal. However, the situation remains mostly obscure for b≥5. In general, the type of results that we obtain present some remarkable similarities with those of the 1990 paper of D. Stanton, who considered the case of partitions inside ordinary (straight) Ferrers shapes. Along the way, we also determine some interesting q-analogs of the binomial coefficients, which in certain instances we conjecture to be unimodal. We state several other conjectures throughout this note, in the hopes to stimulate further work in this area. In particular, one of these will attempt to place into a much broader context the unimodality of the posets M(n) of staircase partitions, for which determining a combinatorial proof remains an outstanding open problem. |
| first_indexed | 2024-09-23T15:38:53Z |
| format | Article |
| id | mit-1721.1/112204 |
| institution | Massachusetts Institute of Technology |
| last_indexed | 2024-09-23T15:38:53Z |
| publishDate | 2017 |
| publisher | Elsevier |
| record_format | dspace |
| spelling | mit-1721.1/1122042022-09-29T15:16:21Z Unimodality of partitions with distinct parts inside Ferrers shapes Zanello, Fabrizio Stanley, Richard P Massachusetts Institute of Technology. Department of Mathematics Stanley, Richard P We investigate the rank-generating function F[subscript λ] of the poset of partitions contained inside a given shifted Ferrers shape λ. When λ has four parts, we show that F [subscript λ] is unimodal when λ=〈n, n-1, n-2, n-3〉, for any n≥4, and that unimodality fails for the doubly-indexed, infinite family of partitions of the form λ=〈n, n-t, n-2t, n-3t〉, for any given t≥2 and n large enough with respect to t.When λ has b≤3 parts, we show that our rank-generating functions F[subscript λ] are all unimodal. However, the situation remains mostly obscure for b≥5. In general, the type of results that we obtain present some remarkable similarities with those of the 1990 paper of D. Stanton, who considered the case of partitions inside ordinary (straight) Ferrers shapes. Along the way, we also determine some interesting q-analogs of the binomial coefficients, which in certain instances we conjecture to be unimodal. We state several other conjectures throughout this note, in the hopes to stimulate further work in this area. In particular, one of these will attempt to place into a much broader context the unimodality of the posets M(n) of staircase partitions, for which determining a combinatorial proof remains an outstanding open problem. National Science Foundation (U.S.) (Grant DMS-1068625) 2017-11-16T16:57:13Z 2017-11-16T16:57:13Z 2015-04 2014-07 2017-10-27T19:26:20Z Article http://purl.org/eprint/type/JournalArticle 0195-6698 1095-9971 http://hdl.handle.net/1721.1/112204 Stanley, Richard P., and Zanello, Fabrizio. “Unimodality of Partitions with Distinct Parts Inside Ferrers Shapes.” European Journal of Combinatorics 49 (October 2015): 194–202 © 2015 Elsevier https://orcid.org/0000-0003-3123-8241 http://dx.doi.org/10.1016/J.EJC.2015.03.007 European Journal of Combinatorics Creative Commons Attribution-NonCommercial-NoDerivs License http://creativecommons.org/licenses/by-nc-nd/4.0/ application/pdf Elsevier MIT Web Domain |
| spellingShingle | Zanello, Fabrizio Stanley, Richard P Unimodality of partitions with distinct parts inside Ferrers shapes |
| title | Unimodality of partitions with distinct parts inside Ferrers shapes |
| title_full | Unimodality of partitions with distinct parts inside Ferrers shapes |
| title_fullStr | Unimodality of partitions with distinct parts inside Ferrers shapes |
| title_full_unstemmed | Unimodality of partitions with distinct parts inside Ferrers shapes |
| title_short | Unimodality of partitions with distinct parts inside Ferrers shapes |
| title_sort | unimodality of partitions with distinct parts inside ferrers shapes |
| url | http://hdl.handle.net/1721.1/112204 https://orcid.org/0000-0003-3123-8241 |
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