TY - JOUR

T1 - Dyck Paths with Peaks Avoiding or Restricted to a Given Set

AU - Eu, Sen Peng

AU - Liu, Shu Chung

AU - Yeh, Yeong Nan

PY - 2003/11

Y1 - 2003/11

N2 - In this paper we focus on Dyck paths with peaks avoiding or restricted to an arbitrary set of heights. The generating functions of such types of Dyck paths can be represented by continued fractions. We also discuss a special case that requires all peak heights to either lie on or avoid a congruence class (or classes) modulo k. The case when k = 2 is especially interesting. The two sequences for this case are proved, combinatorially as well as algebraically, to be the Motzkin numbers and the Riordan numbers. We introduce the concept of shift equivalence on sequences, which in turn induces an equivalence relation on avoiding and restricted sets. Several interesting equivalence classes whose representatives are well-known sequences are given as examples.

AB - In this paper we focus on Dyck paths with peaks avoiding or restricted to an arbitrary set of heights. The generating functions of such types of Dyck paths can be represented by continued fractions. We also discuss a special case that requires all peak heights to either lie on or avoid a congruence class (or classes) modulo k. The case when k = 2 is especially interesting. The two sequences for this case are proved, combinatorially as well as algebraically, to be the Motzkin numbers and the Riordan numbers. We introduce the concept of shift equivalence on sequences, which in turn induces an equivalence relation on avoiding and restricted sets. Several interesting equivalence classes whose representatives are well-known sequences are given as examples.

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U2 - 10.1111/1467-9590.t01-1-00042

DO - 10.1111/1467-9590.t01-1-00042

M3 - Article

AN - SCOPUS:0142215381

SN - 0022-2526

VL - 111

SP - 453

EP - 465

JO - Studies in Applied Mathematics

JF - Studies in Applied Mathematics

IS - 4

ER -