  We study languages and formal power series associated to (variants of)
Hammersley's process. We show that the ordinary Hammersley process yields a
regular language and the Hammersley tree process yields deterministic
context-free (but non-regular) languages. For the extension to intervals of the
Hammersley process we show that there are two relevant formal languages. One of
them leads to the same class of languages as the ordinary Hammersley tree
process. The other one yields non-context-free languages. The results are
motivated by the problem of studying the analog of the famous Ulam-Hammersley
problem for heapable sequences. Towards this goal we also give an algorithm for
computing formal power series associated to the variants of Hammersley's
process. We employ these algorithms to settle the nature of the scaling
constant, conjectured in previous work to be the golden ratio. Our results
provide experimental support to this conjecture.

Formal Languages and Automata Theory;Discrete Mathematics;Combinatorics;Probability