TOPICS
Search

Stack Polyomino

StackPolygon

A stack polyomino is a self-avoiding convex polyomino containing two adjacent corners of its minimal bounding rectangle. The number of stack polyominoes with perimeter 2n+4 is the Fibonacci number F_(2n), having generating function

sum_(n=0)^inftyF_(2n)t^(2n)=(1-t^2)/((1-t-t^2)(1+t-t^2))
(1)

(Delest and Viennot 1984).

The anisotropic area and perimeter generating function G(x,y) and partial generating functions H_m(y), 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)=-y^(2m-3)q^(m^2-2m)H_m(y,q)
(3)

and

G(x,y)+y^3G(x/y^2,1/y)=0
(4)

(Bousquet-Mélou et al. 1999).

See also

Convex Polyomino, Lattice Polygon, Self-Avoiding Polygon

Explore with Wolfram|Alpha

References

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.Delest, M.-P. and Viennot, G. "Algebraic Languages and Polyominoes [sic] Enumeration." Theoret. Comput. Sci. 34, 169-206, 1984.Wright, E. M. "Stacks." Quart. J. Math. (Oxford) 19, 313-320, 1968.

Referenced on Wolfram|Alpha

Stack Polyomino

Cite this as:

Weisstein, Eric W. "Stack Polyomino." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/StackPolyomino.html

Subject classifications