RSK of an adjacent transposition ($(a-1,a)⟷(P,P),\mathrm{shape}P=(n-1,1)$)

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 $1,\dots ,a-2,a,a-1,a+1,\dots ,n$ 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.