Acyclic heaps of pieces, I

Heaps of pieces were introduced by Viennot and have applications to algebraic combinatorics, theoretical computer science and statistical physics. In this paper, we how certain combinatorial properties of heaps studied by Fan and by Stembridge are closely related to the properties of a certain linear map $\partial_E$ associated to a heap $E$. We examine the relationship between $\partial_E$ and $\partial_F$ when $F$ is a subheap of $E$. This approach allows neat statements and proofs of results on certain associative algebras (generalized Temperley--Lieb algebras) that are otherwise tricky to prove. The key to the proof is to interpret the structure constants of the aforementioned algebras in terms of the maps $\partial$.