Given a partition λ and a cell v in its Ferrers diagram, we define the arm, leg, coarm, coleg,
hook and rim hook of v in λ. It is known that the two statistics “hook length” and “part length”
are equidistributied and symmetric over all partitions of n. We construct an involution φ
exchanging “hook length” and “part length” of all partitions of n, which yields two statistics are
symmetric for all partitions of n. For nonnegative integers α, α′, β and β′ satisfying α+α′ = β+β′,
this involution φ makes a new bijection changing arm length α to α′ and leg length β to β′ over
all partitions of n. It follows bijectively that arm length and leg length are super-symmetric.
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