The sharp time decay rate of the isentropic Navier-Stokes system in $ {\mathop{\mathbb R\kern 0pt}\nolimits}^3 $
June 2021, 29(2): 1925-1944. doi: 10.3934/era.2020098
Some multivariate polynomials for doubled permutations
Bin Han
Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel
Received May 2020 Revised August 2020 Published June 2021 Early access September 2020
Fund Project: Supported by the Israel Science Foundation, grant no. 1970/18
Figure(2)
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.
Keywords: Jacobi elliptic function, continued fraction, Gamma positivity, doubled permutation, alternating permutation.
Mathematics Subject Classification: Primary: 05A05, 05A19; Secondary: 05A15.
Citation: Bin Han. Some multivariate polynomials for doubled permutations. Electronic Research Archive, 2021, 29 (2) : 1925-1944. doi: 10.3934/era.2020098
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Figure 1. The doubled path $ \gamma $
Figure 2. MFS-actions on $ 569174328 $ (recall $ \pi(0) = \pi(10) = 0 $)
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