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Recently, there has been a series on articles on the number of fillings of cell diagrams with restrictions on their increasing and decreasing chains, most notably the paper by Chen, Deng, Du, Stanley and Yan on matchings and set partitions with restrictions on their "crossings" and "nestings," the paper by Bousquet-Mélou and Steingrímsson
on involutions with restrictions on their increasing subsequences, and
the paper by Jonsson on fillings of "moon polyominoes" with restrictions
on their increasing chains. In this talk I shall outline that a unifying
way to approach these problems is by the diagrammatic versions of the
Robinson-Schensted algorithm (due to Fomin) and of Sch\"utzenberger's jeu
de taquin (due to van Leeuwen). This covers work by Martin Rubey and work
by myself.
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