Schr\"oder partitions, Schr\"oder tableaux and weak poset patterns

We introduce the notions of Schr\"oder shape and of Schr\"oder tableau, which provide some kind of analogs of the classical notions of Young shape and Young tableau. We investigate some properties of the partial order given by containment of Schr\"oder shapes. Then we propose an algorithm which is the natural analog of the well known RS correspondence for Young tableaux, and we characterize those permutations whose insertion tableaux have some special shapes. The last part of the article relates the notion of Schr\"oder tableau with those of interval order and of weak containment (and strong avoidance) of posets. We end our paper with several suggestions for possible further work.