Rowmotion on fences
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| _version_ | 1866915259923562496 |
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| author | Elizalde, Sergi Plante, Matthew Roby, Tom Sagan, Bruce |
| author_facet | Elizalde, Sergi Plante, Matthew Roby, Tom Sagan, Bruce |
| contents | A fence is a poset with elements F = {x_1, x_2, ..., x_n} and covers x_1 < x_2 < ... < x_a > x_{a+1} > ... > x_b < x_{b+1} < ... where a, b, ... are positive integers. We investigate rowmotion on antichains and ideals of F. In particular, we show that orbits of antichains can be visualized using tilings. This permits us to prove various homomesy results for the number of elements of an antichain or ideal in an orbit. Rowmotion on fences also exhibits a new phenomenon, which we call homometry, where the value of a statistic is constant on orbits of the same size. Along the way, we prove a general homomesy result for all self-dual posets. We end with some conjectures and avenues for future research. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2108_12443 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Rowmotion on fences Elizalde, Sergi Plante, Matthew Roby, Tom Sagan, Bruce Combinatorics Group Theory 05E18 (Primary) 06A07 (Secondary) A fence is a poset with elements F = {x_1, x_2, ..., x_n} and covers x_1 < x_2 < ... < x_a > x_{a+1} > ... > x_b < x_{b+1} < ... where a, b, ... are positive integers. We investigate rowmotion on antichains and ideals of F. In particular, we show that orbits of antichains can be visualized using tilings. This permits us to prove various homomesy results for the number of elements of an antichain or ideal in an orbit. Rowmotion on fences also exhibits a new phenomenon, which we call homometry, where the value of a statistic is constant on orbits of the same size. Along the way, we prove a general homomesy result for all self-dual posets. We end with some conjectures and avenues for future research. |
| title | Rowmotion on fences |
| topic | Combinatorics Group Theory 05E18 (Primary) 06A07 (Secondary) |
| url | https://arxiv.org/abs/2108.12443 |