Toppling on permutations with an extra chip

TL;DR

This paper extends the study of toppling on permutations with an extra chip to all positions and values, providing a full characterization of permutations that topple to the identity and classifying resultant permutations.

Contribution

It offers a complete characterization of all permutations that topple to the identity and classifies resultant permutations in the generalized setting.

Findings

Resultant permutations are certain decomposable permutations.

Number of configurations toppling to a permutation depends on record statistics.

Number of permutations toppling to a permutation relates to binomial transform of poly-Bernoulli numbers.

Abstract

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 $n-p$ and the number of right-to-left minima to the right of $n-p$. 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.