Stationary 1-dependent Counting Processes: from Runs to Bivariate Generating Functions

We give a formula for the bivariate generating function of a stationary 1-dependent counting process in terms of its run probability generating function, with a probabilistic proof. The formula reduces to the well known bivariate generating function of the Eulerian distribution in the case of descents of a sequence of indepependent and identically distributed random variables. The formula is compared with alternative expressions from the theory of determinantal point processes and the combinatorics of sequences.