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Abstract

In arXiv:1808.06095 we have introduced the Knuth class of the word recording a sequence of locations for repeated internal insertion operations in the Sagan-Stanley skew RSK correspondence, with no prescribed external insertion of new cells, to be a preserver for the $P$-tableau. As a consequence the Benkart-Sottile-Stroomer switching involution on ballot tableau pairs allows a realization as a recursive internal insertion procedure. This amounts to explain the various presentations of Littlewood-Richardson (LR) commuters and their coincidence predicted by Pak and Vallejo with contributions by Danilov and Koshevoi. As an instance, the coincidence of LR commuters is instrumental on bijections between branching models.