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We consider paths in the square lattice and use a valuation called the winding number in order to exhibit some combinatorial properties on these paths. As a corollary, we obtain a characteristic property of non-crossing closed paths, generalizing in this way a result of Daurat and Nivat (2003) on the boundary properties of polyominoes concerning salient and reentrant points. Moreover we obtain a similar result for hexagonal lattices and show that there is no other regular lattice having that property.

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No abstract

By using the discrete version of Green's theorem and bivariate difference calculus we provide incremental algorithms to compute various statistics about polyominoes given, as input, by 4-letter words describing their contour. These statistics include area, coordinates of the center of gravity, moment of inertia, higher order moments, size of projections, hook lengths, number of pixels in common with a given set of pixels and also q-statistics.