Stanley’s statement (EC1 p. 61):

Let \(S \subseteq [n-1]\). Then \(\beta _n(S) \leq E_n\), with equality if and only if \(S = \{ 1, 3, 5, \dots \} \cap [n-1]\) or \(S = \{ 2, 4, 6, \dots \} \cap [n-1]\).

Here \(E_n\) is the \(n\)-th Euler number, realized as \(\beta (\mathrm{Alt}_n)\).

Our formalization, in contrapositive form: for \(n \geq 2\) and \(D \subseteq \{ 0, \dots , n-1\} \),

The “\(S = \{ 2, 4, 6, \dots \} \)” alternative in Stanley’s statement reduces to the complementation symmetry already proved at the \(\beta \)-level (, see also 1.15): the two alternating descent sets have equal \(\beta \).