Schensted Algorithms for Dual Graded Graphs
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This paper is a sequel to . We keep the notation and terminology and extend the numbering of sections, propositions, and formulae of .
The main result of this paper is a generalization of the Robinson-Schensted correspondence to the class of dual graded graphs introduced in , This class extends the class of Y-graphs, or differential posets , for which a generalized Schensted correspondence was constructed earlier in .
The main construction leads to unified bijective proofs of various identities related to path counting, including those obtained in . It is also applied to permutation enumeration, including rook placements on Ferrers boards and enumeration of involutions.
As particular cases of the general construction, we re-derive the classical algorithm of Robinson, Schensted, and Knuth [19, 12], the Sagan-Stanley , Sagan-Worley [16, 29] and Haiman's  algorithms and the author's algorithm for the Young-Fibonacci graph . Some new applications are suggested.
The rim hook correspondence of Stanton and White  and Viennot's bijection  are also special cases of the general construction of this paper.
In , the results of this paper and the previous paper  were presented in a form of extended abstract.
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