The study of toppling on permutations with an extra labeled chip was initiated by the first author with D. Hathcock and P. Tetali (arXiv:2010.11236), where the extra chip was added in the middle. We extend this to all possible locations $p$ as well as values $r$ of the extra chip and give a complete characterization of permutations which topple to the identity. Further, we classify all permutations which are outcomes of the toppling process in this generality, which we call resultant permutations. Resultant permutations turn out to be certain decomposable permutations. The number of configurations toppling to a given resultant permutation is shown to depend purely on the number of left-to-right maxima (or records) of the permutation to the left of $p$ and the number of right-to-left minima to the right of $p$. The number of permutations toppling to a given resultant permutation (identity or otherwise) is shown to be the binomial transform of a poly-Bernoulli number of type B.
@article{10_37236_10420,
author = {Arvind Ayyer and Be\'ata B\'enyi},
title = {Toppling on permutations with an extra chip},
journal = {The electronic journal of combinatorics},
year = {2021},
volume = {28},
number = {4},
doi = {10.37236/10420},
zbl = {1478.05002},
url = {http://geodesic.mathdoc.fr/articles/10.37236/10420/}
}
TY - JOUR
AU - Arvind Ayyer
AU - Beáta Bényi
TI - Toppling on permutations with an extra chip
JO - The electronic journal of combinatorics
PY - 2021
VL - 28
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/10420/
DO - 10.37236/10420
ID - 10_37236_10420
ER -
%0 Journal Article
%A Arvind Ayyer
%A Beáta Bényi
%T Toppling on permutations with an extra chip
%J The electronic journal of combinatorics
%D 2021
%V 28
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/10420/
%R 10.37236/10420
%F 10_37236_10420
Arvind Ayyer; Beáta Bényi. Toppling on permutations with an extra chip. The electronic journal of combinatorics, Tome 28 (2021) no. 4. doi: 10.37236/10420
