Admissible adjacent swap of a standard Young tableau ($s_{i}T,|c_T(i+1)-c_T(i)|>1$)

Swapping consecutive labels $i,i+1$ preserves a standard Young tableau exactly when their cells are incomparable in the row-and-column order. Their contents then differ by more than one in absolute value. These admissible swaps connect all standard tableaux of any fixed shape, as adjacent swaps connect linear extensions of a partially ordered set.