4.1 The toggle action
For \(D, E \subseteq \{ 0, \dots , n-1\} \), \(D \, \triangle \, E = (D \setminus E) \cup (E \setminus D)\).
\(D\, \triangle \, \{ i\} = D \, \triangle \, \{ i\} \) flips the membership of position \(i\) in \(D\).
\((D\, \triangle \, \{ i\} )\, \triangle \, \{ i\} = D\).
Stanley’s Fact #2 (EC1 pp. 57–58) describes how the descent set changes under \(\psi _{i}\): in the right-only-child case \(D(\psi _{i}(w)) = D(w) \, \triangle \, \{ i\} \), while in the two-children case \(D(\psi _{i}(w)) = D(w) \, \triangle \, \{ i-1, i\} \).
The combinatorial heart of this — that toggling a single descent at position \(i\) propagates correctly through “blocks” of consecutive descents/ascents — is captured in by \(\texttt{block\_ left\_ le}\), \(\texttt{block\_ right\_ ge}\), \(\texttt{block\_ descent\_ chain}\), \(\texttt{block\_ chain\_ step}\), \(\texttt{block\_ chain\_ values}\) (all \(\rocqok \)).