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Bar Graph Polygon


BarGraphPolygon

A column-convex self-avoiding polygon which contains the bottom edge of its minimal bounding rectangle. The anisotropic perimeter and area generating function

 G(x,y,q)=sum_(m>=1)sum_(n>=1)sum_(a>=a)C(m,n,a)x^my^nq^a,
(1)

where C(m,n,a) is the number of polygons with 2m horizonal bonds, 2n vertical bonds, and area a, has been computed exactly for the bar graph polygons (Bousquet-Mélou 1996, Bousquet-Mélou et al. 1999). The anisotropic area and perimeter generating function G(x,y,q) and partial generating functions H_m(y,q), connected by

 G(x,y,q)=sum_(m>=1)H_m(y,q)x^m,
(2)

satisfy the self-reciprocity and inversion relations

 H_m(1/y,1/q)=((-1)^m)/(yq^m)H_m(y,q)
(3)

and

 G(x,y,q)-yG(-xq,1/y,1/q)=0
(4)

(Bousquet-Mélou et al. 1999).


See also

Lattice Polygon, Self-Avoiding Polygon

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References

Bousquet-Mélou, M. "A Method for Enumeration of Various Classes of Column-Convex Polygons." Disc. Math. 154, 1-25, 1996.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.

Referenced on Wolfram|Alpha

Bar Graph Polygon

Cite this as:

Weisstein, Eric W. "Bar Graph Polygon." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BarGraphPolygon.html

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