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Schensted Algorithms for Dual Graded Graphs

Abstract

This paper is a sequel to [3]. We keep the notation and terminology and extend the numbering of sections, propositions, and formulae of [3].

The main result of this paper is a generalization of the Robinson-Schensted correspondence to the class of dual graded graphs introduced in [3], This class extends the class of Y-graphs, or differential posets [22], for which a generalized Schensted correspondence was constructed earlier in [2].

The main construction leads to unified bijective proofs of various identities related to path counting, including those obtained in [3]. 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 [18], Sagan-Worley [16, 29] and Haiman's [11] algorithms and the author's algorithm for the Young-Fibonacci graph [2]. Some new applications are suggested.

The rim hook correspondence of Stanton and White [23] and Viennot's bijection [28] are also special cases of the general construction of this paper.

In [5], the results of this paper and the previous paper [3] were presented in a form of extended abstract.

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Fomin, S. Schensted Algorithms for Dual Graded Graphs. Journal of Algebraic Combinatorics 4, 5–45 (1995). https://doi.org/10.1023/A:1022404807578

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  • discrete algorithm
  • enumerative combinatorics
  • poset
  • Young diagram