## Seungsang Oh, Enumeration of multiple self-avoiding polygons in a confined square lattice

Enumeration of multiple self-avoiding polygons in a confined square lattice
2014/10/07 *Tuesday 5PM-6PM*
Room 1409
$$\mathbb{Z}_{m \times n}$$ is the rectangle of size $$m \times n$$ in the square lattice. $$p_{m \times n}$$ denotes the cardinality of multiple self-avoiding polygons in $$\mathbb{Z}_{m \times n}$$.

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 $$O_1 = \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right)$$.

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