The limiting distribution of the coefficients of the $q$-Catalan numbers
HTML articles powered by AMS MathViewer
- by William Y. C. Chen, Carol J. Wang and Larry X. W. Wang PDF
- Proc. Amer. Math. Soc. 136 (2008), 3759-3767 Request permission
Abstract:
We show that the limiting distributions of the coefficients of the $q$-Catalan numbers and the generalized $q$-Catalan numbers are normal. Despite the fact that these coefficients are not unimodal for small $n$, we conjecture that for sufficiently large $n$, the coefficients are unimodal and even log-concave except for a few terms of the head and tail.References
- Horst Alzer, Sharp bounds for the Bernoulli numbers, Arch. Math. (Basel) 74 (2000), no. 3, 207–211. MR 1739499, DOI 10.1007/s000130050432
- George E. Andrews, Catalan numbers, $q$-Catalan numbers and hypergeometric series, J. Combin. Theory Ser. A 44 (1987), no. 2, 267–273. MR 879684, DOI 10.1016/0097-3165(87)90033-1
- J. Fürlinger and J. Hofbauer, $q$-Catalan numbers, J. Combin. Theory Ser. A 40 (1985), no. 2, 248–264. MR 814413, DOI 10.1016/0097-3165(85)90089-5
- Ira Gessel and Dennis Stanton, Applications of $q$-Lagrange inversion to basic hypergeometric series, Trans. Amer. Math. Soc. 277 (1983), no. 1, 173–201. MR 690047, DOI 10.1090/S0002-9947-1983-0690047-7
- Christian Krattenthaler, A new $q$-Lagrange formula and some applications, Proc. Amer. Math. Soc. 90 (1984), no. 2, 338–344. MR 727262, DOI 10.1090/S0002-9939-1984-0727262-6
- Guy Louchard and Helmut Prodinger, The number of inversions in permutations: a saddle point approach, J. Integer Seq. 6 (2003), no. 2, Article 03.2.8, 19. MR 1998753
- Barbara H. Margolius, Permutations with inversions, J. Integer Seq. 4 (2001), no. 2, Article 01.2.4, 13. MR 1873402
- Vladimir N. Sachkov, Probabilistic methods in combinatorial analysis, Encyclopedia of Mathematics and its Applications, vol. 56, Cambridge University Press, Cambridge, 1997. Translated from the Russian; Revised by the author. MR 1453118, DOI 10.1017/CBO9780511666193
- Richard P. Stanley, Log-concave and unimodal sequences in algebra, combinatorics, and geometry, Graph theory and its applications: East and West (Jinan, 1986) Ann. New York Acad. Sci., vol. 576, New York Acad. Sci., New York, 1989, pp. 500–535. MR 1110850, DOI 10.1111/j.1749-6632.1989.tb16434.x
- Richard P. Stanley, Enumerative combinatorics. Vol. 1, Cambridge Studies in Advanced Mathematics, vol. 49, Cambridge University Press, Cambridge, 1997. With a foreword by Gian-Carlo Rota; Corrected reprint of the 1986 original. MR 1442260, DOI 10.1017/CBO9780511805967
Additional Information
- William Y. C. Chen
- Affiliation: Center for Combinatorics, LPMC-TJKLC, Nankai University, Tianjin 300071, People’s Republic of China
- MR Author ID: 232802
- Email: chen@nankai.edu.cn
- Carol J. Wang
- Affiliation: Center for Combinatorics, LPMC-TJKLC, Nankai University, Tianjin 300071, People’s Republic of China
- Email: wangjian@cfc.nankai.edu.cn
- Larry X. W. Wang
- Affiliation: Center for Combinatorics, LPMC-TJKLC, Nankai University, Tianjin 300071, People’s Republic of China
- MR Author ID: 845775
- Email: wxw@cfc.nankai.edu.cn
- Received by editor(s): August 20, 2007
- Published electronically: July 3, 2008
- Additional Notes: The authors are grateful to the referee for valuable suggestions. Thanks are also due to Barbara Margolius and Helmut Prodinger for helpful comments. This work was supported by the 973 Project, the PCSIRT Project of the Ministry of Education, the Ministry of Science and Technology, and the National Science Foundation of China.
- Communicated by: Jim Haglund
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 136 (2008), 3759-3767
- MSC (2000): Primary 05A16, 60C05
- DOI: https://doi.org/10.1090/S0002-9939-08-09464-1
- MathSciNet review: 2425713