A convex polyomino (sometimes called a "convex polygon") is a polyomino whose perimeter is equal to that of its minimal bounding box (Bousquet-Mélou et al. 1999). Furthermore, if it contains at least one corner of its minimal bounding box, it is said to be a directed convex polyomino. A column-convex polyomino is a self-avoiding polyomino such that the intersection of any vertical line with the polyomino has at most two connected components, and a row-convex polyomino is similarly defined.
Klarner and Rivest (1974) and Bender (1974) gave the asymptotic estimate for the number 
of convex polyominoes having area 
as
 |
(1)
|
with 
and 
(Delest and Viennot 1984).
The anisotropic perimeter and area generating function
 |
(2)
|
where 
is the number of polygons with 
horizonal bonds, 
vertical bonds, and area 
is given by
![G(x,y,q)=2sum_(m>=1)(y^(m+2))/((xq)_m^2N(xq^(m-1))N(xq^m))[T_(m+1)S(xq^m)-yT_mS(xq^(m+1))]^2+sum_(m>=1)(xy^mq^m(T_m)^2)/((xq)_(m-1)(xq)_m),](/images/equations/ConvexPolyomino/NumberedEquation3.svg) |
(3)
|
where
 |  |  |
(4)
|
 |  | ![sum_(n>=1)[(x^nq^n)/((yq)_n)sum_(j=0)^(n-1)((-1)^jq^((j; 2)))/((q)_j(yq^(j+1))_(n-j))]](/images/equations/ConvexPolyomino/Inline14.svg) |
(5)
|
and 
is the polynomial recurrence relation
 |
(6)
|
with 
and 
(Bousquet-Mélou 1992b). The first few of these polynomials are given by
 |  |  |
(7)
|
 |  |  |
(8)
|
 |  |  |
(9)
|
 |  |  |
(10)
|
Expanding the generating function shows that the number of convex polyominoes having perimeter 
is given by
 |
(11)
|
where 
,
, and 
is a binomial coefficient
(Delest and Viennot 1984, Bousquet-Mélou 1992ab). The generating function
for 
is explicitly given by
 |
(12)
|
(Delest and Viennot 1984; Guttmann and Enting 1988). The first few terms are therefore 1, 2, 7, 28, 120, 528, 2344, 10416, ... (OEIS A005436).
This function has been computed exactly for the column-convex and directed column-convex polyominoes (Bousquet-Mélou 1996, Bousquet-Mélou et al. 1999).
is a q-series,
but becomes algebraic for column-convex polyominoes. However, 
for column-convex polyominoes again involves q-series
(Temperley 1956, Bousquet-Mélou et al. 1999).
is an algebraic function
of 
and 
(called the "fugacities") given by
 |  |  |
(13)
|
 |  | ![(R(x,y)xy)/([Delta(x,y)]^2)-(4x^2y^2)/(Delta^(3/2)),](/images/equations/ConvexPolyomino/Inline45.svg) |
(14)
|
where
 |  |  |
(15)
|
 |  |  |
(16)
|
 |  | ![(1-y)^2[1-(x(2+2y-x))/((1-y)^2)]](/images/equations/ConvexPolyomino/Inline54.svg) |
(17)
|
(Lin and Chang 1988, Bousquet-Mélou 1992ab). This can be solved to explicitly give
 |
(18)
|
(Gessel 2000, Bousquet-Mélou 1992ab).
satisfies the inversion relation
 |
(19)
|
where
 |  |  |
(20)
|
 |  | ![(1-y)^2[1-(x(2+2y-x))/((1-y)^2)]](/images/equations/ConvexPolyomino/Inline61.svg) |
(21)
|
(Lin and Chang 1988, Bousquet-Mélou et al. 1999).
The half-vertical perimeter and area generating function for column-convex polyominos of width 3 is given by the special case
 |
(22)
|
of the general rational function (Bousquet-Mélou et al. 1999), which satisfies the reciprocity relation
 |
(23)
|
The anisotropic area and perimeter generating function 
and partial generating functions 
, connected by
 |
(24)
|
satisfy the self-reciprocity and inversion relations
 |
(25)
|
and
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(26)
|
(Bousquet-Mélou et al. 1999).
-ominoes." Disc. Math. 8, 31-40, 1974.Bousquet-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. "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. 
-ominoes." Disc. Math. 8, 31-40, 1974.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.Sloane, N. J. A. Sequence