The transfer matrices and the capacity of the 2-dimensional (1, ∞)-runlength limited constraint.

A 2-dimensional (1,∞)-runlength limited constraint Shs is the set of (0,1)-matrices/arrays that do not have adjacent horizontal or vertical 1’s. Let f(m,n) be the number of (m+1)×(n+1) (m,n≥0) matrices that satisfy the constraint Shs. It is known (Calkin and Wilf, 1998) that f(m,n) is equal to the number of independent sets in an (m+1)×(n+1) grid graph and f(m,n)=1tTmn1, where Tm is an associated transfer matrix and 1 is the all 1’s vector. The capacity of the constraint Shs, cap(Shs)=limm,n→+∞log2f(m,n)mn, has been bounded actively in the last decades (e.g., Calkin and Wilf, 1998; Engel, 1990; Nagy and Zeger, 2000; Weber, 1988), with the most recent bounds being 0.5878911617...≤cap(Shs)≤0.5878911618... (Nagy and Zeger, 2000).