4.2 The \(\omega (D)\) map
For \(D \subseteq \{ 0, \dots , n\} \), the omega set is
\[ \omega (D) \; =\; \{ \, k \in \{ 0, \dots , n-1\} : (k \in D) \oplus (k+1 \in D)\, \} \; \subseteq \; \{ 0, \dots , n-1\} . \]
This is exactly Stanley’s \(\omega (S)\) from EC1 just before eq. (1.64), p. 60: positions \(k\) where exactly one of \(\{ k, k+1\} \) lies in \(D\). Equivalently, \(\omega (D) = D \, \triangle \, (D - 1)\) where \(D - 1\) is \(D\) shifted by one.
\(k \in \omega (D)\) iff toggling \(D\) at one of \(\{ k, k+1\} \) flips the membership at the other position.