학술논문
Counting Berg partitions via Sturmian words and substitution tilings
Document Type
Working Paper
Source
Subject
Language
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}.
Comment: 14 pages, 3 figures
Comment: 14 pages, 3 figures