Counting Berg partitions via Sturmian words and substitution tilings
Artur Siemaszko, Maciej P. Wojtkowski · 2013
We develop the connection of Berg partitions with special substitution tilings of two tiles. We obtain a new proof that the number of Berg partitions with a fi xed connectivity matrix is equal to half of the sum of its entries, \[12]. This approach together with the formula of Seebold \[10], for the number of substitutions preserving a given Sturmian sequence, shows that all of the combinatorial substitutions can be realized geometrically as Berg partitions. We treat Sturmian tilings as intersection tilings of bi-partitions. Using the symmetries of bi-partitions we obtain geometrically the palindromic properties of Sturmian sequences (Theorem 3) established combinatorially by de Luca and Mignosi, \[6].