We develop a particle cluster technique for representing the equilibrium thermodynamics of an inhomogeneous multispecies classical fluid in one dimension with nearest-neighbor interactions. The corresponding free energy functionals take on the general Wertheim association form and make contact with the Widom insertion argument. Application is made to adhesive rod mixtures, for which first-order thermodynamic perturbation theory is seen to be exact, and to more general interactions, such as the square well, in which activity is no longer a local functional of the density and explicit relations must be generated by recursion. For a fluid on a loop, auxiliary two-point feedback functions appear, which correct the simply connected form.