  The partition function of the two-dimensional Ising model on a square lattice
with nearest-neighbour interactions and periodic boundary conditions is
investigated. Kaufman [Phys. Rev. 76, 1232--1243 (1949)] gave a solution for
this function consisting of four summands. The summands are rewritten as
functions of a low-temperature expansion variable, resulting in polynomials
with integer coefficients. Considering these polynomials for system sizes
$2^m\times 2^n$ ($m,n\in\N$), a variety of recursion relations in $m,n$ are
found. The recursions reveal a rich structure of the partition function and can
be employed to render the computer algebra calculation of the microcanonical
partition function more efficient.
