4.4 The alternating descent set and equation (1.65)
\()\mathrm{Alt}_n = \{ \, i \in \{ 0, \dots , n-1\} : i \text{ is odd}\, \} \). Stanley’s “alternating set” is \(S = \{ 1, 3, 5, \dots \} \cap [n-1]\) (eq. 1.65, p. 60); via our index shift this matches \(\mathrm{Alt}_n\) exactly.
\(D \subseteq \{ 0, \dots , n-1\} \) is set-alternating iff for every consecutive pair \(i, i+1\) in \(\{ 0, \dots , n-1\} \), exactly one of \(\{ i, i+1\} \) belongs to \(D\). There are exactly two such sets on \(\{ 0, \dots , n-1\} \) (the “odd-position” and “even-position” alternating descents).
For \(m \geq 1\), \(\omega ((\mathrm{Alt}_{m+2})) \; =\; \{ 0, \dots , m\} \; =\; \mathrm{set\_ T}.\) This is the “\(\Leftarrow \)” direction of Stanley’s eq. (1.65) (p. 60): the alternating descent set has full omega.
For \(D \subseteq \{ 0, \dots , m\} \) with \(D\) not set-alternating, \(\omega (D) \subsetneq \mathrm{set\_ T}\).
This is the contrapositive of the “\(\Rightarrow \)” direction of Stanley’s eq. (1.65): non-alternating sets have strictly smaller omega. Together with 4.12 it gives the full characterization