  We introduce two stationary versions of two discrete variants of Hammersley's
process in a finite box, this allows us to recover in a unified and simple way
the laws of large numbers proved by T. Sepp{\"a}l{\"a}inen for two generalized
Ulam's problems. As a by-product we obtain an elementary solution for the
original Ulam problem. We also prove that for the first process defined on Z,
Bernoulli product measures are the only extremal and translation-invariant
stationary measures.
