Jonathan Bloom ; Dan Saracino - Modified Growth Diagrams, Permutation Pivots, and the BWX Map $\phi^*$

dmtcs:3018 - Discrete Mathematics & Theoretical Computer Science, January 1, 2012, DMTCS Proceedings vol. AR, 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012) - https://doi.org/10.46298/dmtcs.3018
Modified Growth Diagrams, Permutation Pivots, and the BWX Map $\phi^*$

Authors: Jonathan Bloom ; Dan Saracino

    In their paper on Wilf-equivalence for singleton classes, Backelin, West, and Xin introduced a transformation $\phi^*$, defined by an iterative process and operating on (all) full rook placements on Ferrers boards. Bousquet-Mélou and Steingrimsson proved the analogue of the main result of Backelin, West, and Xin in the context of involutions, and in so doing they needed to prove that $\phi^*$ commutes with the operation of taking inverses. The proof of this commutation result was long and difficult, and Bousquet-Mélou and Steingrimsson asked if $\phi^*$ might be reformulated in such a way as to make this result obvious. In the present paper we provide such a reformulation of $\phi^*$, by modifying the growth diagram algorithm of Fomin. This also answers a question of Krattenthaler, who noted that a bijection defined by the unmodified Fomin algorithm obviously commutes with inverses, and asked what the connection is between this bijection and $\phi^*$.


    Volume: DMTCS Proceedings vol. AR, 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012)
    Section: Proceedings
    Published on: January 1, 2012
    Imported on: January 31, 2017
    Keywords: Wilf-equivalence, RSK correspondence, Growth Diagrams, Bijection, Permutations,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

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