Abstract
The contact map of a protein fold in the two-dimensional (2D) square lattice has arc length at least 3, and each internal vertex has degree at most 2, whereas the two terminal vertices have degree at most 3. Recently, Chen, Guo, Sun, and Wang studied the enumeration of
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$$\it m$$
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-regular linear stacks, where each arc has length at least
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$$\it m$$
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and the degree of each vertex is bounded by 2. Since the two terminal points in a protein fold in the 2D square lattice may form contacts with at most three adjacent lattice points, we are led to the study of extended
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$$\it m$$
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-regular linear stacks, in which the degree of each terminal point is bounded by 3. This model is closed to real protein contact maps. Denote the generating functions of the
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$$\it m$$
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-regular linear stacks and the extended
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$$\it m$$
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-regular linear stacks by
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$$\it {R_m} ( x )$$
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and
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$$\it R_m^{ [ 3 ] } ( x )$$
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, respectively. We show that
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$$\it R_m^{ [ 3 ] } ( x )$$
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can be written as a rational function of
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$$\it {R_m} ( x )$$
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. For a certain
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, by eliminating
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$$\it {R_m} ( x )$$
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, we obtain an equation satisfied by
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$$\it R_m^{ [ 3 ] } ( x )$$
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and derive the asymptotic formula of the numbers of
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$$\it m$$
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-regular linear stacks of length
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$$\it n$$
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.
