Korea University

Room 1409

In this talk, we construct an algorithm producing the precise value of \(p_{m \times n}\) for positive integers m,n that uses recurrence relations of state matrices which turn out to be remarkably efficient to count such polygons. $$ p_{m \times n} = \mbox{(1,1)-entry of the matrix } (X_m)^n -1$$ where the matrix \(X_m\) is defined by $$ X_{k+1} = \left( \begin{array}{cc} X_k & O_k \\ O_k & X_k \end{array}\right) \ \ \mbox{and} \ \ \ O_{k+1} = \left( \begin{array}{cc} O_k & X_k \\ X_k & 0 \end{array} \right) $$ for \(k=1, \cdots, m-1\), with \(1 \times 1\) matrices \(X_1 = \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right)\) and \(O_1 = \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right)\).