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LARGE SCHRÖDER PATHS BY TYPES AND SYMMETRIC FUNCTIONS
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 Title & Authors
LARGE SCHRÖDER PATHS BY TYPES AND SYMMETRIC FUNCTIONS
An, Su Hyung; Eu, Sen-Peng; Kim, Sangwook;
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 Abstract
In this paper we provide three results involving large Schrder paths. First, we enumerate the number of large Schrder 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 Schrder paths. These results extend their counterparts of Kreweras and Armstrong-Eu on Dyck paths respectively.
 Keywords
Schrder paths;partial horizontal strips;sparse noncrossing partitions;elementary symmetric functions;
 Language
English
 Cited by
 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. crossref(new window)

3.
N. Dershowitz and S. Zaks, Enumerations of ordered trees, Discrete Math. 31 (1980), no. 1, 9-28. crossref(new window)

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. crossref(new window)

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. crossref(new window)

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. crossref(new window)

9.
G. Kreweras, Sur les partitions non croisees d'un cycle, Discrete Math. 1 (1972), no. 4, 333-350. crossref(new window)

10.
T. S. Nanjundiah, Remark on a note of P. Turan, Amer. Math. Monthly 65 (1958), 354. crossref(new window)

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. crossref(new window)

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. crossref(new window)