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Schensted algorithms for dual graded graphs. (English) Zbl 0817.05077
From the author’s abstract: This paper is a sequel to the author’s paper [Duality of graded graphs, J. Algebr. Comb. 3, No. 4, 357-404 (1994; Zbl 0810.05005)]. The main result is a generalization of the Robinson- Schensted correspondence to the class of dual graded graphs. This class extends the class of \(Y\)-graphs, or differential posets, for which a generalized Schensted correspondence was constructed earlier. The main construction leads to unified bijective proofs of various identities related to path counting. It is also applied to permutation enumeration, including rook placements on Ferrers boards and enumeration of involutions.
As particular cases of the general construction the classical algorithm of Robinson, Schensted, and Knuth, the Sagan-Stanley, Sagan-Worley and Haiman’s algorithms and the author’s algorithm for the Young-Fibonacci graph are re-derived. Some new application 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.
Reviewer: K.Engel (Rostock)

05E10 Combinatorial aspects of representation theory
05A15 Exact enumeration problems, generating functions
06A07 Combinatorics of partially ordered sets
Full Text: DOI
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