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  • Seungsang Oh, Enumeration of multiple self-avoiding polygons in a confined square lattice
    Enumeration of multiple self-avoiding polygons in a confined square lattice
    Seungsang Oh (오승상)
    Korea University
    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}. </p>

    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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