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We extend the transfer matrix method of one-dimensional hard core fluidsplaced between confining walls for that case where the particles can pass eachother and at most two layers can form. We derive an eigenvalue equation for aquasi-one-dimensional system of hard squares confined between two parallelwalls, where the pore width is between $\sigma$ and $3\sigma$ ( $\sigma$ is theside length of the square). The exact equation of state and the nearestneighbour distribution functions show three different structures: a fluid phasewith one layer, a fluid phase with two layers and a solid-like structure wherethe fluid layers are strongly correlated. The structural transition betweendifferently ordered fluids develops continuously with increasing density, i.e.no thermodynamic phase transition occurs. The high density structure of thesystem consists of clusters with two layers which are broken with particlesstaying in the middle of the pore.

Beyond the single-file fluid limit using transfer matrix method: Exact results for confined parallel hard squares