  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.
