Define the minimal bounding rectangle as the smallest rectangle containing a given lattice polygon. If the perimeter of the lattice polygon is equal to that of its
minimal bounding rectangle, it is said to be convex. (Note that a "convex"
lattice polygon is not necessarily convex in the usual sense of the word.) A staircase
polygon is then defined as a convex polygon which contains two opposite corners of
its bounding rectangle (Bousquet-Mélou et al. 1999).

The area generating function that counts polygons of width for staircase polygons of width 4 is given by

(1)

which satisfies

(2)

(Bousquet-Mélou 1992, Bousquet-Mélou et al. 1999). The anisotropic area and perimeter generating function and partial generating functions , connected by

(3)

satisfy the self-reciprocity and inversion relations

(4)

for
and

(5)

(Bousquet-Mélou et al. 1999).

The anisotropic area and perimeter generating function of staircase polygon with a staircase hole satisfies
an inversion relation of the form

(6)

(Bousquet-Mélou et al. 1999).

Knuth (2022) considered the packing of all staircase polygons with a given semiperimeter into a square.