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In this paper we provide three results involving large Schr¨oder paths. First, we enumerate the number of large Schr¨oder 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¨oder paths. These results extend their counterparts of Kreweras and Armstrong-Eu on Dyck paths respectively.
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참고문헌 (14)
참고문헌 신청
A. M. Garsia / 1996 / A remarkable q, t-Catalan sequence and q-Lagrange inversion / J. Algebraic Combin 5 (3) : 191 ~ 244
D. Armstrong / 2008 / Nonhomogeneous parking functions and noncrossing partitions / Electron. J. Combin 15 (1) : 12 ~
E. S. Egge / 2003 / A Schr¨oder generalization of Haglund’s statistic on Catalan paths / Electron. J. Combin 10 : 21 ~
G. Kreweras / 1972 / Sur les partitions non crois´ees d’un cycle / Discrete Math 1 (4) : 333 ~ 350
J. Bandlow / 2002 / A weight-preserving bijection between Schr¨oder paths and Schr¨oder permutations / Ann. Comb 6 (3-4) : 235 ~ 248
D. Armstrong / 2008 / Nonhomogeneous parking functions and noncrossing partitions / Electron. J. Combin 15 (1) : 12 ~
E. S. Egge / 2003 / A Schr¨oder generalization of Haglund’s statistic on Catalan paths / Electron. J. Combin 10 : 21 ~
G. Kreweras / 1972 / Sur les partitions non crois´ees d’un cycle / Discrete Math 1 (4) : 333 ~ 350
J. Bandlow / 2002 / A weight-preserving bijection between Schr¨oder paths and Schr¨oder permutations / Ann. Comb 6 (3-4) : 235 ~ 248
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