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)
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(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.
