We study a birational map associated to any finite poset $P$. This map is a far-reaching generalization (found by Einstein and Propp) of classical rowmotion, which is a certain permutation of the set of order ideals of $P$. Classical rowmotion has been studied by various authors (Fon-der-Flaass, Cameron, Brouwer, Schrijver, Striker, Williams and many more) under different guises (Striker-Williams promotion and Panyushev complementation are two examples of maps equivalent to it). In contrast, birational rowmotion is new and has yet to reveal several of its mysteries. In this paper, we set up the tools for analyzing the properties of iterates of this map, and prove that it has finite order for a certain class of posets which we call "skeletal". Roughly speaking, these are graded posets constructed from one-element posets by repeated disjoint union and "grafting onto an antichain"; in particular, any forest having its leaves all on the same rank is such a poset. We also make a parallel analysis of classical rowmotion on this kind of posets, and prove that the order in this case equals the order of birational rowmotion.
@article{10_37236_4334,
author = {Darij Grinberg and Tom Roby},
title = {Iterative properties of birational rowmotion. {I:} {Generalities} and skeletal posets.},
journal = {The electronic journal of combinatorics},
year = {2016},
volume = {23},
number = {1},
doi = {10.37236/4334},
zbl = {1338.06003},
url = {http://geodesic.mathdoc.fr/articles/10.37236/4334/}
}
TY - JOUR
AU - Darij Grinberg
AU - Tom Roby
TI - Iterative properties of birational rowmotion. I: Generalities and skeletal posets.
JO - The electronic journal of combinatorics
PY - 2016
VL - 23
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/4334/
DO - 10.37236/4334
ID - 10_37236_4334
ER -
%0 Journal Article
%A Darij Grinberg
%A Tom Roby
%T Iterative properties of birational rowmotion. I: Generalities and skeletal posets.
%J The electronic journal of combinatorics
%D 2016
%V 23
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/4334/
%R 10.37236/4334
%F 10_37236_4334
Darij Grinberg; Tom Roby. Iterative properties of birational rowmotion. I: Generalities and skeletal posets.. The electronic journal of combinatorics, Tome 23 (2016) no. 1. doi: 10.37236/4334
