3 The cd-index and Stanley’s Fact #3

This chapter formalizes Stanley EC1 §1.6.3, pp. 57–60: the cd-letter classification of internal vertices, the cd-monomial \(\Phi _{w}\), the characteristic monomial \(u_{S}\) of a descent set, and the multiset identity (Fact #3, p. 58)

The substantive support theorem \(\Phi _{w} = \sum _{\omega (X) \supseteq S_w} u_{X}\) (Stanley’s identity inside the proof of Prop 1.6.4, p. 60–61) is proved here as .