 Solution

Use the convention that insertion returns the insertion tableau $P$ first and the recording tableau $Q$ second. Successive row insertions of $3,4,1,6,7,5,2$ giveFor example inserting five bumps six to the second row, and the final insertion of two bumps four from the first row and six from the second, creating the third row. The newly created boxes at these steps receive recording labels six and seven respectively.

The deletion input is this pair with the two tableaux interchanged. The symmetry property of RSK therefore gives the inverse permutation. One can also reverse-insert the boxes in descending recording-label order. The result isIndeed the displayed second row lists the positions of one through seven in the original permutation.