The transfer matrix: design your rule
Choose which lines may sit together, and count
Choose which line adjacencies you allow and look at the matrix, its powers, the counts and its dominant eigenvalue.
The Fibonacci in the hexagram experiment was one case: forbidding two adjacent yin. Here you choose which adjacencies you allow, with a 2x2 matrix M where M[a][b] = 1 if line b may sit above a. Its powers count the figures by number of lines (1ᵀMⁿ⁻¹1), its trace the cyclic ones, and its dominant eigenvalue is the growth ratio. It is the z-transform of the count; Laplace's does not apply, because the I Ching has no continuous time: the honest no beside the discrete yes.
Outside the adjacency rules there is another famous count hidden: the balanced hexagrams (3 yang, 3 yin) where, reading from the bottom up, yang never falls behind yin. They are the Dyck paths, and there are exactly C3 = 5.
the 5 Catalan hexagrams: 42, 44, 50, 52, 56
Every adjacency rule between lines is a 2x2 matrix; its powers give the counts by number of lines, its trace the cyclic ones and its dominant eigenvalue the growth ratio (phi for the Fibonacci rules, 2 for the free one, 1 for alternation).
verificar_transferenciaThe balanced hexagrams where yang never falls behind yin are exactly C₃ = 5, the values 42, 44, 50, 52 and 56 (Catalan numbers): it is the z-transform of the count, not Laplace’s, because there is no continuous time.
verificar_transferenciaThe types of claim and the full bibliography (APA) are in Foundations.