= RSK of an adjacent transposition
{title2=$(a-1,a)\longleftrightarrow(P,P),\quad\operatorname{shape}P=(n-1,1)$}
If a <standard Young tableau> of shape $(n-1,1)$ has second-row entry $a$, then its self-pair under <RSK> corresponds to the <adjacent transposition> $(a-1,a)$. Inserting the word $1,\ldots,a-2,a,a-1,a+1,\ldots,n$ first appends increasing entries to the first row. At time $a$, inserting $a-1$ bumps $a$ into the unique second-row cell. Every later entry appends to the first row. Thus the insertion tableau is the stated tableau, and the recording tableau has its second-row label $a$ as well. This proves the correspondence directly, including $n=2$, without invoking a separate theorem about <involutions>.
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