Journal of Algebraic Combinatorics

, Volume 4, Issue 1, pp 5–45

Schensted Algorithms for Dual Graded Graphs

  • Sergey Fomin
Article

DOI: 10.1023/A:1022404807578

Cite this article as:
Fomin, S. Journal of Algebraic Combinatorics (1995) 4: 5. doi:10.1023/A:1022404807578

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.

discrete algorithmenumerative combinatoricsposetYoung diagram

Copyright information

© Kluwer Academic Publishers 1995

Authors and Affiliations

  • Sergey Fomin
    • 1
    • 2
  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridge
  2. 2.Theory of Algorithms Laboratory, SPIIRANRussian Academy of SciencesRussia