A convex polyomino containing at least one edge of its minimal bounding rectangle. The perimeter and area generating function for directed polygons of width , height , and area is given by
where
(Bousquet-Mélou 1992ab).
The anisotropic perimeter generating function for directed convex polygons of width and height is given by
where
(Lin and Chang 1988, Bousquet 1992ab, Bousquet-Mélou et al. 1999). This can be solved to explicitly give
(10)
(Bousquet-Mélou 1992ab). Expanding the generating function gives
An explicit formula of is given by Bousquet-Mélou (1992ab). These functions satisfy the reciprocity relations
(14)
(15)
(Bousquet-Mélou et al. 1999).
The anisotropic area and horizontal perimeter generating function and partial generating functions , connected by
(16)
satisfy the self-reciprocity and inversion relations
(17)
and
(18)
(Bousquet-Mélou et al. 1999).
See alsoConvex Polyomino,
Lattice Polygon Explore with Wolfram|Alpha ReferencesBousquet-Mélou, M. "Convex Polyominoes and Heaps of Segments." J. Phys. A: Math. Gen. 25, 1925-1934, 1992a.Bousquet-Mélou, M. "Convex Polyominoes and Algebraic Languages." J. Phys. A: Math. Gen. 25, 1935-1944, 1992b.Bousquet-Mélou, M.; Guttmann, A. J.; Orrick, W. P.; and Rechnitzer, A. "Inversion Relations, Reciprocity and Polyominoes." 23 Aug 1999. http://arxiv.org/abs/math.CO/9908123.Lin, K. Y. and Chang, S. J. "Rigorous Results for the Number of Convex Polygons on the Square and Honeycomb Lattices." J. Phys. A: Math. Gen. 21, 2635-2642, 1988. Referenced on Wolfram|AlphaDirected Convex Polyomino Cite this as:Weisstein, Eric W. "Directed Convex Polyomino." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/DirectedConvexPolyomino.html
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