DOI QR코드

DOI QR Code

LARGE SCHRÖDER PATHS BY TYPES AND SYMMETRIC FUNCTIONS

  • An, Su Hyung (Department of Mathematics Yonsei University) ;
  • Eu, Sen-Peng (Department of Mathematics National Taiwan Normal University) ;
  • Kim, Sangwook (Department of Mathematics Chonnam National University)
  • Received : 2013.11.19
  • Published : 2014.07.31

Abstract

In this paper we provide three results involving large Schr$\ddot{o}$der paths. First, we enumerate the number of large Schr$\ddot{o}$der paths by type. Second, we prove that these numbers are the coefficients of a certain symmetric function defined on the staircase skew shape when expanded in elementary symmetric functions. Finally we define a symmetric function on a Fuss path associated with its low valleys and prove that when expanded in elementary symmetric functions the indices are running over the types of all Schr$\ddot{o}$der paths. These results extend their counterparts of Kreweras and Armstrong-Eu on Dyck paths respectively.

Keywords

References

  1. D. Armstrong and S.-P. Eu, Nonhomogeneous parking functions and noncrossing partitions, Electron. J. Combin. 15 (2008), no. 1, Research Paper 146, 12 pp.
  2. J. Bandlow, E. S. Egge, and K. Killpatrick, A weight-preserving bijection between Schroder paths and Schroder permutations, Ann. Comb. 6 (2002), no. 3-4, 235-248. https://doi.org/10.1007/s000260200000
  3. N. Dershowitz and S. Zaks, Enumerations of ordered trees, Discrete Math. 31 (1980), no. 1, 9-28. https://doi.org/10.1016/0012-365X(80)90168-5
  4. E. S. Egge, J. Haglund, K. Killpatrick, and D. Kremer, A Schroder generalization of Haglund's statistic on Catalan paths, Electron. J. Combin. 10 (2003), Research Paper 16, 21 pp.
  5. S.-P. Eu and T.-S. Fu, Lattice paths and generalized cluster complexes, J. Combin. Theory Ser. A 115 (2008), no. 7, 1183-1210. https://doi.org/10.1016/j.jcta.2007.12.011
  6. A. M. Garsia and M. Haiman, A remarkable q, t-Catalan sequence and q-Lagrange inversion, J. Algebraic Combin. 5 (1996), no. 3, 191-244. https://doi.org/10.1023/A:1022476211638
  7. J. Haglund, A proof of the q, t-Schroder conjecture, Int. Math. Res. Not. (2004), no. 11, 525-560.
  8. M. D. Haiman, Conjectures on the quotient ring by diagonal invariants, J. Algebraic Combin. 3 (1994), no. 1. 17-76. https://doi.org/10.1023/A:1022450120589
  9. G. Kreweras, Sur les partitions non croisees d'un cycle, Discrete Math. 1 (1972), no. 4, 333-350. https://doi.org/10.1016/0012-365X(72)90041-6
  10. T. S. Nanjundiah, Remark on a note of P. Turan, Amer. Math. Monthly 65 (1958), 354. https://doi.org/10.2307/2308802
  11. J. H. Przytycki and A. S. Sikora, Polygon dissections and Euler, Fuss, Kirkman, and Cayley Numbers, J. Combin. Theory Ser. A 92 (2000), no. 1, 68-76. https://doi.org/10.1006/jcta.1999.3042
  12. R. P. Stanley, Parking functions and noncrossing partitions, The Wilf Festschrift (Philadelphia, PA, 1996), Electron. J. Combin. 4 (1997), no. 1, Research Paper 20, 14 pp.
  13. R. P. Stanley, Enumerative combinatorics. Vol. 2, volume 62 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1999.
  14. J. West, Generating trees and the Catalan and Schroder numbers, Discrete Math. 146 (1995), no. 1-3, 247-262. https://doi.org/10.1016/0012-365X(94)00067-1