  The exact transfer-matrix solution for the longitudinal equilibrium
properties of the single-file hard-disk fluid is used to study the limiting
low- and high-pressure behaviors analytically as functions of the pore width.
In the low-pressure regime, the exact third and fourth virial coefficients are
obtained, which involve single and double integrals, respectively. Moreover, we
show that the standard irreducible diagrams do not provide a complete account
of the virial coefficients in confined geometries. The asymptotic equation of
state in the high-pressure limit is seen to present a simple pole at the
close-packing linear density, as in the hard-rod fluid, but, in contrast to the
latter case, the residue is $2$. Since, for an arbitrary pressure, the exact
transfer-matrix treatment requires the numerical solution of an eigenvalue
integral equation, we propose here two simple approximations to the equation of
state, with different complexity levels, and carry out an extensive assessment
of their validity and practical convenience vs the exact solution and available
computer simulations.
