Rowmotion on fences

Elizalde, Sergi; Plante, Matthew; Roby, Tom; Sagan, Bruce E.

doi:10.5802/alco.256

Rowmotion on fences

Elizalde, Sergi ¹; Plante, Matthew ²; Roby, Tom ²; Sagan, Bruce E.³

¹ Department of Mathematics Dartmouth College Hanover NH 03755
² Department of Mathematics University of Connecticut Storrs CT 06269
³ Department of Mathematics Michigan State University East Lansing MI 48824

Algebraic Combinatorics, Volume 6 (2023) no. 1, pp. 17-36.

Abstract

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} ⊲ \dots

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.

Received: 2021-08-27
Revised: 2022-06-28
Accepted: 2022-07-07
Published online: 2023-02-24

DOI: 10.5802/alco.256

Classification: 05E18, 06A07
Keywords: fence poset, homomesy, homometry, rowmotion, tiling

Author's affiliations:

Elizalde, Sergi ¹; Plante, Matthew ²; Roby, Tom ²; Sagan, Bruce E. ³

¹ Department of Mathematics Dartmouth College Hanover NH 03755
² Department of Mathematics University of Connecticut Storrs CT 06269
³ Department of Mathematics Michigan State University East Lansing MI 48824

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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Elizalde, Sergi; Plante, Matthew; Roby, Tom; Sagan, Bruce E. Rowmotion on fences. Algebraic Combinatorics, Volume 6 (2023) no. 1, pp. 17-36. doi : 10.5802/alco.256. https://alco.centre-mersenne.org/articles/10.5802/alco.256/

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