Abstract:

Recently, there has been a series of papers 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 Steíngrimsson 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 Schützenberger's jeu de taquin (due to van Leeuwen). This covers work by Martin Rubey and work by myself.