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Some multivariate polynomials for doubled permutations

  • Received: 01 May 2020 Revised: 01 August 2020 Published: 16 September 2020
  • Primary: 05A05, 05A19; Secondary: 05A15

  • Flajolet and Françon [European. J. Combin. 10 (1989) 235-241] gave a combinatorial interpretation for the Taylor coefficients of the Jacobian elliptic functions in terms of doubled permutations. We show that a multivariable counting of the doubled permutations has also an explicit continued fraction expansion generalizing the continued fraction expansions of Rogers and Stieltjes. The second goal of this paper is to study the expansion of the Taylor coefficients of the generalized Jacobian elliptic functions, which implies the symmetric and unimodal property of the Taylor coefficients of the generalized Jacobian elliptic functions. The main tools are the combinatorial theory of continued fractions due to Flajolet and bijections due to Françon-Viennot, Foata-Zeilberger and Clarke-Steingrímsson-Zeng.

    Citation: Bin Han. Some multivariate polynomials for doubled permutations[J]. Electronic Research Archive, 2021, 29(2): 1925-1944. doi: 10.3934/era.2020098

    Related Papers:

  • Flajolet and Françon [European. J. Combin. 10 (1989) 235-241] gave a combinatorial interpretation for the Taylor coefficients of the Jacobian elliptic functions in terms of doubled permutations. We show that a multivariable counting of the doubled permutations has also an explicit continued fraction expansion generalizing the continued fraction expansions of Rogers and Stieltjes. The second goal of this paper is to study the expansion of the Taylor coefficients of the generalized Jacobian elliptic functions, which implies the symmetric and unimodal property of the Taylor coefficients of the generalized Jacobian elliptic functions. The main tools are the combinatorial theory of continued fractions due to Flajolet and bijections due to Françon-Viennot, Foata-Zeilberger and Clarke-Steingrímsson-Zeng.



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