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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  September 2020

Fund Project: Supported by the Israel Science Foundation, grant no. 1970/18

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
References:
[1]

C. A. Athanasiadis, Gamma-positivity in combinatorics and geometry, Sém. Lothar. Combin., 77 (2016-2018), 64pp.  Google Scholar

[2]

F. Bowman, Introduction to Elliptic Functions with Applications, English Universities Press, Ltd., London, 1953.  Google Scholar

[3]

P. Brändén, Actions on permutations and unimodality of descent polynomials, European J. Combin., 29 (2008), 514-531.  doi: 10.1016/j.ejc.2006.12.010.  Google Scholar

[4]

R. J. Clarke, E. Steingrímsson and J. Zeng, New Euler-Mahonian statistics on permutations and words, Adv. in Appl. Math., 18 (1997), 237-270.  doi: 10.1006/aama.1996.0506.  Google Scholar

[5]

E. V. F. Conrad, Some Continued Fraction Expansions of Laplace Transforms of Elliptic Functions., Ph.D thesis, Ohio State University, 2002.  Google Scholar

[6]

E. V. F. Conrad and P. Flajolet, The Fermat cubic, elliptic functions, continued fractions and a combinatorial excursion, Sém. Lothar. Combin., 54 (2005/07), 44pp.  Google Scholar

[7]

S. Corteel, Crossings and alignments of permutations, Adv. in Appl. Math., 38 (2007) 149–163. doi: 10.1016/j.aam.2006.01.006.  Google Scholar

[8]

D. Dumont, A combinatorial interpretation for the Schett recurrence on the Jacobian elliptic functions, Math. Comp., 33 (1979), 1293-1297.  doi: 10.1090/S0025-5718-1979-0537974-1.  Google Scholar

[9]

D. Dumont, Une approche combinatoire des fonctions elliptiques de Jacobi, Adv. in Math., 41 (1981), 1-39.  doi: 10.1016/S0001-8708(81)80002-3.  Google Scholar

[10]

D. Dumont, Pics de cycle et dérivées partielles, Sém. Lothar. Combin., 13 (1986), 19pp. Google Scholar

[11]

P. Flajolet, Combinatorial aspects of continued fractions, Discrete Math., 32 (1980), 125-161.  doi: 10.1016/0012-365X(80)90050-3.  Google Scholar

[12]

P. Flajolet and J. Françon, Elliptic functions, continued fractions and doubled permutations, European J. Combin., 10 (1989), 235-241.  doi: 10.1016/S0195-6698(89)80057-5.  Google Scholar

[13]

D. Foata and M.-P. Schützenberger, Théorie Géométrique des Polynômes Eulériens, Lecture Notes in Mathematics, 138, Springer-Verlag, Berlin-New York, 1970. doi: 10.1007/BFb0060799.  Google Scholar

[14]

D. Foata and V. Strehl, Rearrangements of the symmetric group and enumerative properties of the tangent and secant numbers, Math Z., 137 (1974), 257-264.  doi: 10.1007/BF01237393.  Google Scholar

[15]

D. Foata and D. Zeilberger, Denert's permutation statistic is indeed Euler-Mahonian, Stud. Appl. Math., 83 (1990), 31-59.  doi: 10.1002/sapm199083131.  Google Scholar

[16]

B. Han, J. Mao and J. Zeng, Eulerian polynomials and excedance statistics, Adv. in Appl. Math., 121 (2020). doi: 10.1016/j.aam.2020.102092.  Google Scholar

[17]

B. Han, J. Mao and J. Zeng, Eulerian polynomials and excedance statistics via continued fractions, Sém. Lothar. Combin., 84B (2020), 12pp. Google Scholar

[18]

Z. Lin and J. Zeng, The $\gamma$-positivity of basic Eulerian polynomials via group actions, J. Combin. Theory Ser. A., 135 (2015), 112-129.  doi: 10.1016/j.jcta.2015.04.006.  Google Scholar

[19]

S.-M. Ma, T. Mansour, D. G. L. Wang and Y.-N. Yeh, Several variants of the Dumont differential system and permutation statistics, Sci. China Math., 62 (2019), 2033-2052.  doi: 10.1007/s11425-016-9240-5.  Google Scholar

[20]

S.-M. Ma, J. Ma, Y.-N. Yeh and R. R. Zhou, On the unimodality of the Taylor expansion coefficients of Jacobian Elliptic function, preprint, arXiv: 1807.08700v3. Google Scholar

[21]

L. J. Rogers, on the representation of certain asymptotic series as convergent continued fractions, Proc. London Math. Soc. (2), 4 (1907), 72-89.  doi: 10.1112/plms/s2-4.1.72.  Google Scholar

[22]

H. Shin and J. Zeng, The $q$-tangent and $q$-secant numbers via continued fractions, European J. Combin., 31 (2010), 1689-1705.  doi: 10.1016/j.ejc.2010.04.003.  Google Scholar

[23]

H. Shin and J. Zeng, The symmetric and unimodal expansion of Eulerian polynomials via continued fractions, European J. Combin., 33 (2012), 111-127.  doi: 10.1016/j.ejc.2011.08.005.  Google Scholar

[24]

H. Shin and J. Zeng, Symmetric unimodal expansions of excedances in colored permutations, European J. Combin., 52 (2016), 174-196.  doi: 10.1016/j.ejc.2015.10.004.  Google Scholar

[25]

A. D. Sokal and J. Zeng, Some multivariate master polynomials for permutations, set partitions, and perfect matchings, and their continued fractions, preprint, arXiv: 2003.08192. Google Scholar

[26]

R. P. Stanley, A survey of alternating permutations, in Combinatorics and Graphs, Contemp. Math., 531, Amer. Math. Soc., Providence, RI, 2010,165–196. doi: 10.1090/conm/531/10466.  Google Scholar

[27]

T.-J. Stieltjes, Sur the réduction en fraction continue d'une série procédant suivant les puissances descendantes d'une variable, Ann. Fac. Sci. Tulouse Sci. Math. Sci. Phys., 3 (1889), H1–H17. doi: 10.5802/afst.34.  Google Scholar

[28]

G. Viennot, Une interprétation combinatoire des coefficients de déveloooements en série entière des fonctions elliptiques de Jacobi, J. Combin. Theory Ser. A, 29 (1980), 121-133.  doi: 10.1016/0097-3165(80)90001-1.  Google Scholar

[29]

S. H. F. Yan, H. Zhou and Z. Lin, A new encoding of permutations by Laguerre histories, Electron. J. Combin., 26 (2019), 9pp. doi: 10.37236/8661.  Google Scholar

show all references

References:
[1]

C. A. Athanasiadis, Gamma-positivity in combinatorics and geometry, Sém. Lothar. Combin., 77 (2016-2018), 64pp.  Google Scholar

[2]

F. Bowman, Introduction to Elliptic Functions with Applications, English Universities Press, Ltd., London, 1953.  Google Scholar

[3]

P. Brändén, Actions on permutations and unimodality of descent polynomials, European J. Combin., 29 (2008), 514-531.  doi: 10.1016/j.ejc.2006.12.010.  Google Scholar

[4]

R. J. Clarke, E. Steingrímsson and J. Zeng, New Euler-Mahonian statistics on permutations and words, Adv. in Appl. Math., 18 (1997), 237-270.  doi: 10.1006/aama.1996.0506.  Google Scholar

[5]

E. V. F. Conrad, Some Continued Fraction Expansions of Laplace Transforms of Elliptic Functions., Ph.D thesis, Ohio State University, 2002.  Google Scholar

[6]

E. V. F. Conrad and P. Flajolet, The Fermat cubic, elliptic functions, continued fractions and a combinatorial excursion, Sém. Lothar. Combin., 54 (2005/07), 44pp.  Google Scholar

[7]

S. Corteel, Crossings and alignments of permutations, Adv. in Appl. Math., 38 (2007) 149–163. doi: 10.1016/j.aam.2006.01.006.  Google Scholar

[8]

D. Dumont, A combinatorial interpretation for the Schett recurrence on the Jacobian elliptic functions, Math. Comp., 33 (1979), 1293-1297.  doi: 10.1090/S0025-5718-1979-0537974-1.  Google Scholar

[9]

D. Dumont, Une approche combinatoire des fonctions elliptiques de Jacobi, Adv. in Math., 41 (1981), 1-39.  doi: 10.1016/S0001-8708(81)80002-3.  Google Scholar

[10]

D. Dumont, Pics de cycle et dérivées partielles, Sém. Lothar. Combin., 13 (1986), 19pp. Google Scholar

[11]

P. Flajolet, Combinatorial aspects of continued fractions, Discrete Math., 32 (1980), 125-161.  doi: 10.1016/0012-365X(80)90050-3.  Google Scholar

[12]

P. Flajolet and J. Françon, Elliptic functions, continued fractions and doubled permutations, European J. Combin., 10 (1989), 235-241.  doi: 10.1016/S0195-6698(89)80057-5.  Google Scholar

[13]

D. Foata and M.-P. Schützenberger, Théorie Géométrique des Polynômes Eulériens, Lecture Notes in Mathematics, 138, Springer-Verlag, Berlin-New York, 1970. doi: 10.1007/BFb0060799.  Google Scholar

[14]

D. Foata and V. Strehl, Rearrangements of the symmetric group and enumerative properties of the tangent and secant numbers, Math Z., 137 (1974), 257-264.  doi: 10.1007/BF01237393.  Google Scholar

[15]

D. Foata and D. Zeilberger, Denert's permutation statistic is indeed Euler-Mahonian, Stud. Appl. Math., 83 (1990), 31-59.  doi: 10.1002/sapm199083131.  Google Scholar

[16]

B. Han, J. Mao and J. Zeng, Eulerian polynomials and excedance statistics, Adv. in Appl. Math., 121 (2020). doi: 10.1016/j.aam.2020.102092.  Google Scholar

[17]

B. Han, J. Mao and J. Zeng, Eulerian polynomials and excedance statistics via continued fractions, Sém. Lothar. Combin., 84B (2020), 12pp. Google Scholar

[18]

Z. Lin and J. Zeng, The $\gamma$-positivity of basic Eulerian polynomials via group actions, J. Combin. Theory Ser. A., 135 (2015), 112-129.  doi: 10.1016/j.jcta.2015.04.006.  Google Scholar

[19]

S.-M. Ma, T. Mansour, D. G. L. Wang and Y.-N. Yeh, Several variants of the Dumont differential system and permutation statistics, Sci. China Math., 62 (2019), 2033-2052.  doi: 10.1007/s11425-016-9240-5.  Google Scholar

[20]

S.-M. Ma, J. Ma, Y.-N. Yeh and R. R. Zhou, On the unimodality of the Taylor expansion coefficients of Jacobian Elliptic function, preprint, arXiv: 1807.08700v3. Google Scholar

[21]

L. J. Rogers, on the representation of certain asymptotic series as convergent continued fractions, Proc. London Math. Soc. (2), 4 (1907), 72-89.  doi: 10.1112/plms/s2-4.1.72.  Google Scholar

[22]

H. Shin and J. Zeng, The $q$-tangent and $q$-secant numbers via continued fractions, European J. Combin., 31 (2010), 1689-1705.  doi: 10.1016/j.ejc.2010.04.003.  Google Scholar

[23]

H. Shin and J. Zeng, The symmetric and unimodal expansion of Eulerian polynomials via continued fractions, European J. Combin., 33 (2012), 111-127.  doi: 10.1016/j.ejc.2011.08.005.  Google Scholar

[24]

H. Shin and J. Zeng, Symmetric unimodal expansions of excedances in colored permutations, European J. Combin., 52 (2016), 174-196.  doi: 10.1016/j.ejc.2015.10.004.  Google Scholar

[25]

A. D. Sokal and J. Zeng, Some multivariate master polynomials for permutations, set partitions, and perfect matchings, and their continued fractions, preprint, arXiv: 2003.08192. Google Scholar

[26]

R. P. Stanley, A survey of alternating permutations, in Combinatorics and Graphs, Contemp. Math., 531, Amer. Math. Soc., Providence, RI, 2010,165–196. doi: 10.1090/conm/531/10466.  Google Scholar

[27]

T.-J. Stieltjes, Sur the réduction en fraction continue d'une série procédant suivant les puissances descendantes d'une variable, Ann. Fac. Sci. Tulouse Sci. Math. Sci. Phys., 3 (1889), H1–H17. doi: 10.5802/afst.34.  Google Scholar

[28]

G. Viennot, Une interprétation combinatoire des coefficients de déveloooements en série entière des fonctions elliptiques de Jacobi, J. Combin. Theory Ser. A, 29 (1980), 121-133.  doi: 10.1016/0097-3165(80)90001-1.  Google Scholar

[29]

S. H. F. Yan, H. Zhou and Z. Lin, A new encoding of permutations by Laguerre histories, Electron. J. Combin., 26 (2019), 9pp. doi: 10.37236/8661.  Google Scholar

Figure 1.  The doubled path $ \gamma $
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Figure 2.  MFS-actions on $ 569174328 $ (recall $ \pi(0) = \pi(10) = 0 $)
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