Schröder partitions, Schröder tableaux and weak poset patterns

We introduce the notions of Schröder shapes and Schröder tableaux, which provide an analog of the classical notions of Young shapes and Young tableaux. We investigate some properties of the partial order given by containment of Schröder shapes. Then we propose an algorithm that 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 the Schröder tableau with those of interval order and weak containment (and strong avoidance) of posets. We end our paper with several suggestions for possible further work