Counting Berg partitions via Sturmian words and substitution tilings

TL;DR

This paper links Berg partitions with substitution tilings and Sturmian sequences, providing new geometric proofs and revealing symmetries and properties of Sturmian tilings.

Contribution

It introduces a geometric approach to counting Berg partitions and demonstrates how all combinatorial substitutions can be realized as Berg partitions.

Findings

Number of Berg partitions equals half the sum of the connectivity matrix entries.

All combinatorial substitutions can be realized as Berg partitions.

Palindromic properties of Sturmian sequences are derived geometrically.

Abstract

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 fixed connectivity matrix is equal to half of the sum of its entries, \cite{S-W}. This approach together with the formula of S\'{e}\'{e}bold \cite{Seb}, 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, \cite{L-M}.