A convex polyomino (sometimes called a "convex polygon") is a polyomino whose perimeter is equal to that of its minimal bounding box (BousquetMélou 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 columnconvex polyomino is a selfavoiding polyomino such that the intersection of any vertical line with the polyomino has at most two connected components, and a rowconvex 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
(3)

where
(4)
 
(5)

and is the polynomial recurrence relation
(6)

with and (BousquetMélou 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, BousquetMélou 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 columnconvex and directed columnconvex polyominoes (BousquetMélou 1996, BousquetMélou et al. 1999). is a qseries, but becomes algebraic for columnconvex polyominoes. However, for columnconvex polyominoes again involves qseries (Temperley 1956, BousquetMélou et al. 1999).
is an algebraic function of and (called the "fugacities") given by
(13)
 
(14)

where
(15)
 
(16)
 
(17)

(Lin and Chang 1988, BousquetMélou 1992ab). This can be solved to explicitly give
(18)

(Gessel 2000, BousquetMélou 1992ab).
satisfies the inversion relation
(19)

where
(20)
 
(21)

(Lin and Chang 1988, BousquetMélou et al. 1999).
The halfvertical perimeter and area generating function for columnconvex polyominos of width 3 is given by the special case
(22)

of the general rational function (BousquetMélou 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 selfreciprocity and inversion relations
(25)

and
(26)

(BousquetMélou et al. 1999).