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Some multivariate polynomials for doubled permutations
Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel |
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.
References:
[1] |
C. A. Athanasiadis, Gamma-positivity in combinatorics and geometry, Sém. Lothar. Combin., 77 (2016-2018), 64pp. |
[2] |
F. Bowman, Introduction to Elliptic Functions with Applications, English Universities Press, Ltd., London, 1953. |
[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. |
[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. |
[5] |
E. V. F. Conrad, Some Continued Fraction Expansions of Laplace Transforms of Elliptic Functions., Ph.D thesis, Ohio State University, 2002. |
[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. |
[7] |
S. Corteel, Crossings and alignments of permutations, Adv. in Appl. Math., 38 (2007) 149–163.
doi: 10.1016/j.aam.2006.01.006. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
show all references
References:
[1] |
C. A. Athanasiadis, Gamma-positivity in combinatorics and geometry, Sém. Lothar. Combin., 77 (2016-2018), 64pp. |
[2] |
F. Bowman, Introduction to Elliptic Functions with Applications, English Universities Press, Ltd., London, 1953. |
[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. |
[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. |
[5] |
E. V. F. Conrad, Some Continued Fraction Expansions of Laplace Transforms of Elliptic Functions., Ph.D thesis, Ohio State University, 2002. |
[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. |
[7] |
S. Corteel, Crossings and alignments of permutations, Adv. in Appl. Math., 38 (2007) 149–163.
doi: 10.1016/j.aam.2006.01.006. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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