Simultaneous Core Partitions: Parameterizations and Sums
Fix coprime s; t > 1. We re-prove, without Ehrhart reciprocity, a conjecture of Armstrong (recently verified by Johnson) that the definitely many simultaneous (s; t)- cores have average size 1 24 (s - 1)(t - 1)(s+t+1), and that the subset of self-conjugate cores has the same average (first shown...
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| Format: | Article |
| Language: | en_US |
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European Mathematical Information Service (EMIS)
2016
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| Online Access: | http://hdl.handle.net/1721.1/103038 |
| _version_ | 1826198788705878016 |
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| author | Wang, Victor Y. |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Wang, Victor Y. |
| author_sort | Wang, Victor Y. |
| collection | MIT |
| description | Fix coprime s; t > 1. We re-prove, without Ehrhart reciprocity, a conjecture of Armstrong (recently verified by Johnson) that the definitely many simultaneous (s; t)- cores have average size 1 24 (s - 1)(t - 1)(s+t+1), and that the subset of self-conjugate cores has the same average (first shown by Chen{Huang{Wang). We similarly prove a recent conjecture of Fayers that the average weighted by an inverse stabilizer| giving the \expected size of the t-core of a random s-core"|is 1 24 (s - 1)(t2 - 1). We also prove Fayers' conjecture that the analogous self-conjugate average is the same if t is odd, but instead 1 24 (s - 1)(t2 + 2) if t is even. In principle, our explicit methods|or implicit variants thereof|extend to averages of arbitrary powers. The main new observation is that the stabilizers appearing in Fayers' conjectures have simple formulas in Johnson's z-coordinates parameterization of (s; t)-cores. We also observe that the z-coordinates extend to parameterize general t-cores. As an example application with t := s+d, we count the number of (s; s+d; s+2d)- cores for coprime s; d > 1, verifying a recent conjecture of Amdeberhan and Leven. |
| first_indexed | 2024-09-23T11:10:01Z |
| format | Article |
| id | mit-1721.1/103038 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T11:10:01Z |
| publishDate | 2016 |
| publisher | European Mathematical Information Service (EMIS) |
| record_format | dspace |
| spelling | mit-1721.1/1030382022-10-01T01:45:39Z Simultaneous Core Partitions: Parameterizations and Sums Wang, Victor Y. Massachusetts Institute of Technology. Department of Mathematics Wang, Victor Y. Fix coprime s; t > 1. We re-prove, without Ehrhart reciprocity, a conjecture of Armstrong (recently verified by Johnson) that the definitely many simultaneous (s; t)- cores have average size 1 24 (s - 1)(t - 1)(s+t+1), and that the subset of self-conjugate cores has the same average (first shown by Chen{Huang{Wang). We similarly prove a recent conjecture of Fayers that the average weighted by an inverse stabilizer| giving the \expected size of the t-core of a random s-core"|is 1 24 (s - 1)(t2 - 1). We also prove Fayers' conjecture that the analogous self-conjugate average is the same if t is odd, but instead 1 24 (s - 1)(t2 + 2) if t is even. In principle, our explicit methods|or implicit variants thereof|extend to averages of arbitrary powers. The main new observation is that the stabilizers appearing in Fayers' conjectures have simple formulas in Johnson's z-coordinates parameterization of (s; t)-cores. We also observe that the z-coordinates extend to parameterize general t-cores. As an example application with t := s+d, we count the number of (s; s+d; s+2d)- cores for coprime s; d > 1, verifying a recent conjecture of Amdeberhan and Leven. 2016-06-07T15:51:10Z 2016-06-07T15:51:10Z 2016-01 2015-08 Article http://purl.org/eprint/type/JournalArticle 1097-1440 1077-8926 http://hdl.handle.net/1721.1/103038 Wang, Victor Y. "Simultaneous Core Partitions: Parameterizations and Sums." Electronic Journal of Combinatorics 23(1) (2016), p.1-4. en_US http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i1p4 Electronic Journal of Combinatorics Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. application/pdf European Mathematical Information Service (EMIS) European Mathematical Information Service (EMIS) |
| spellingShingle | Wang, Victor Y. Simultaneous Core Partitions: Parameterizations and Sums |
| title | Simultaneous Core Partitions: Parameterizations and Sums |
| title_full | Simultaneous Core Partitions: Parameterizations and Sums |
| title_fullStr | Simultaneous Core Partitions: Parameterizations and Sums |
| title_full_unstemmed | Simultaneous Core Partitions: Parameterizations and Sums |
| title_short | Simultaneous Core Partitions: Parameterizations and Sums |
| title_sort | simultaneous core partitions parameterizations and sums |
| url | http://hdl.handle.net/1721.1/103038 |
| work_keys_str_mv | AT wangvictory simultaneouscorepartitionsparameterizationsandsums |