  A polyomino is a finite, edge-connected set of cells in the plane. At the
present time, an enumeration of all polyominoes is nowhere in sight. On the
other hand, there are several subsets of polyominoes for which generating
functions are known. For example, there exists extensive knowledge about
column-convex polyominoes, a model introduced by Temperley in 1956. While
studying column-convex polyominoes, researchers also gave a look at diagonally
convex polyominoes (DCPs), but noticed an awkward feature: when the last
diagonal of a DCP is deleted, the remaining object is not always a polyomino.
So researchers focused their attention on directed DCPs. (A directed DCP is
such a DCP that remains a polyomino when, for any $i$, its last $i$ diagonals
are deleted.) Directed DCPs gradually became well understood, whereas general
DCPs have remained unexplored up to now.
  In this paper, we finally face general DCPs. Modulo a little trick, which
saves us from dealing with non-polyominoes, we use the layered approach
(described in chapter 3 of the book ``Polygons, Polyominoes and Polycubes",
edited by Anthony Guttmann). The computations are of remarkable bulk. Our main
result is the perimeter generating function for DCPs; we denote it $D(d,x)$.
The function $D(d,x)$ is algebraic and satisfies an equation of degree eight.
The formula for $D(d,x)$ is about eight pages long. That formula involves nine
polynomials in $d$ and $x$, and each of those polynomials is of degree $58$ or
more in $x$. The interested reader can view the formula for $D(d,x)$ in the
Maple worksheet attached to this paper.
