Toppling on Permutations with an Extra Chip

The study of toppling on permutations with an extra labeled chip was initiated by the first author D. Hathcock and P. Tetali (arXiv:2010.11236), where added in middle. We extend this to all possible locations $p$ as well values $r$ give a complete characterization which topple identity. Further, we classify are outcomes process generality, call resultant permutations. Resultant turn out be certain decomposable number configurations given permutation is shown depend purely left-to-right maxima (or records) left $n-p$ right-to-left minima right $n-p$. (identity or otherwise) binomial transform poly-Bernoulli type B.