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

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.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.256
Classification: 05E18, 06A07
Keywords: fence poset, homomesy, homometry, rowmotion, tiling

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
@article{ALCO_2023__6_1_17_0,
     author = {Elizalde, Sergi and Plante, Matthew and Roby, Tom and Sagan, Bruce E.},
     title = {Rowmotion on fences},
     journal = {Algebraic Combinatorics},
     pages = {17--36},
     publisher = {The Combinatorics Consortium},
     volume = {6},
     number = {1},
     year = {2023},
     doi = {10.5802/alco.256},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.256/}
}
TY  - JOUR
AU  - Elizalde, Sergi
AU  - Plante, Matthew
AU  - Roby, Tom
AU  - Sagan, Bruce E.
TI  - Rowmotion on fences
JO  - Algebraic Combinatorics
PY  - 2023
SP  - 17
EP  - 36
VL  - 6
IS  - 1
PB  - The Combinatorics Consortium
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.256/
DO  - 10.5802/alco.256
LA  - en
ID  - ALCO_2023__6_1_17_0
ER  - 
%0 Journal Article
%A Elizalde, Sergi
%A Plante, Matthew
%A Roby, Tom
%A Sagan, Bruce E.
%T Rowmotion on fences
%J Algebraic Combinatorics
%D 2023
%P 17-36
%V 6
%N 1
%I The Combinatorics Consortium
%U https://alco.centre-mersenne.org/articles/10.5802/alco.256/
%R 10.5802/alco.256
%G en
%F ALCO_2023__6_1_17_0
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/

[1] Chan, Melody; Haddadan, Shahrzad; Hopkins, Sam; Moci, Luca The expected jaggedness of order ideals, Forum Math. Sigma, Volume 5 (2017), Paper no. e9, 27 pages | DOI | MR | Zbl

[2] Claussen, Andrew Expansion posets for polygon cluster algebras, Ph. D. Thesis, Michigan State University (2020)

[3] Defant, Colin; Lin, James Rowmotion on m-Tamari and BiCambrian Lattices (In preparation)

[4] Dilks, Kevin; Pechenik, Oliver; Striker, Jessica Resonance in orbits of plane partitions and increasing tableaux, J. Combin. Theory Ser. A, Volume 148 (2017), pp. 244-274 | DOI | MR | Zbl

[5] Einstein, David; Propp, James Piecewise-linear and birational toggling, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014) (Discrete Math. Theor. Comput. Sci. Proc., AT), Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2014, pp. 513-524 | MR | Zbl

[6] Einstein, David; Propp, James Combinatorial, piecewise-linear, and birational homomesy for products of two chains, Algebr. Comb., Volume 4 (2021) no. 2, pp. 201-224 | DOI | Numdam | MR | Zbl

[7] Elizalde, Sergi; Sagan, Bruce E. Partial rank symmetry and fences (2022) | arXiv

[8] Galashin, Pavel; Pylyavskyy, Pavlo R-systems, Selecta Math. (N.S.), Volume 25 (2019) no. 2, Paper no. 22, 63 pages | DOI | MR | Zbl

[9] Gansner, Emden R. On the lattice of order ideals of an up-down poset, Discrete Math., Volume 39 (1982) no. 2, pp. 113-122 | DOI | MR | Zbl

[10] Grinberg, Darij; Roby, Tom Iterative properties of birational rowmotion II: rectangles and triangles, Electron. J. Combin., Volume 22 (2015) no. 3, Paper no. 3.40, 49 pages | MR | Zbl

[11] Grinberg, Darij; Roby, Tom Iterative properties of birational rowmotion I: generalities and skeletal posets, Electron. J. Combin., Volume 23 (2016) no. 1, Paper no. 1.33, 40 pages | MR | Zbl

[12] Joseph, Michael Antichain toggling and rowmotion, Electron. J. Combin., Volume 26 (2019) no. 1, Paper no. 1.29, 43 pages | MR | Zbl

[13] McConville, Thomas; Sagan, Bruce E.; Smyth, Clifford On a rank-unimodality conjecture of Morier-Genoud and Ovsienko, Discrete Math., Volume 344 (2021) no. 8, p. 112483, 13 pages | DOI | MR | Zbl

[14] Morier-Genoud, Sophie; Ovsienko, Valentin q-deformed rationals and q-continued fractions, Forum Math. Sigma, Volume 8 (2020), Paper no. e13, 55 pages | DOI | MR | Zbl

[15] Munarini, Emanuele; Zagaglia Salvi, Norma On the rank polynomial of the lattice of order ideals of fences and crowns, Discrete Math., Volume 259 (2002) no. 1-3, pp. 163-177 | DOI | MR | Zbl

[16] Musiker, Gregg; Roby, Tom Paths to understanding birational rowmotion on products of two chains, Algebr. Comb., Volume 2 (2019) no. 2, pp. 275-304 | DOI | Numdam | MR | Zbl

[17] Musiker, Gregg; Schiffler, Ralf; Williams, Lauren Positivity for cluster algebras from surfaces, Adv. Math., Volume 227 (2011) no. 6, pp. 2241-2308 | DOI | MR | Zbl

[18] Oğuz, Ezgi Kantarcı; Ravichandran, Mohan Rank Polynomials of Fence Posets are Unimodal (2021) | arXiv | DOI

[19] Propp, James The combinatorics of frieze patterns and Markoff numbers, Integers, Volume 20 (2020), Paper no. A12, 38 pages | MR | Zbl

[20] Roby, Tom Dynamical algebraic combinatorics and the homomesy phenomenon, Recent trends in combinatorics (IMA Vol. Math. Appl.), Volume 159, Springer, [Cham], 2016, pp. 619-652 | DOI | MR | Zbl

[21] Schiffler, Ralf A cluster expansion formula (A n case), Electron. J. Combin., Volume 15 (2008) no. 1, Paper no. 64, 9 pages | MR | Zbl

[22] Schiffler, Ralf On cluster algebras arising from unpunctured surfaces. II, Adv. Math., Volume 223 (2010) no. 6, pp. 1885-1923 | DOI | MR | Zbl

[23] Schiffler, Ralf; Thomas, Hugh On cluster algebras arising from unpunctured surfaces, Int. Math. Res. Not. IMRN (2009) no. 17, pp. 3160-3189 | DOI | MR | Zbl

[24] Striker, Jessica Dynamical algebraic combinatorics: promotion, rowmotion, and resonance, Notices Amer. Math. Soc., Volume 64 (2017) no. 6, pp. 543-549 | DOI | MR | Zbl

[25] Striker, Jessica Rowmotion and generalized toggle groups, Discrete Math. Theor. Comput. Sci., Volume 20 (2018) no. 1, Paper no. 17, 26 pages | MR | Zbl

[26] Striker, Jessica; Williams, Nathan Promotion and rowmotion, European J. Combin., Volume 33 (2012) no. 8, pp. 1919-1942 | DOI | MR | Zbl

[27] Thomas, Hugh; Williams, Nathan Rowmotion in slow motion, Proc. Lond. Math. Soc. (3), Volume 119 (2019) no. 5, pp. 1149-1178 | DOI | MR | Zbl

[28] Vorland, Corey Homomesy in products of three chains and multidimensional recombination, Electron. J. Combin., Volume 26 (2019) no. 4, Paper no. 4.30, 26 pages | MR | Zbl

[29] Yurikusa, Toshiya Cluster expansion formulas in type A, Algebr. Represent. Theory, Volume 22 (2019) no. 1, pp. 1-19 | DOI | MR | Zbl

[30] Yurikusa, Toshiya Combinatorial cluster expansion formulas from triangulated surfaces, Electron. J. Combin., Volume 26 (2019) no. 2, Paper no. 2.33, 39 pages | MR | Zbl

Cited by Sources: