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تسجيل مستخدم جديدPotential Polynomials and Motzkin Paths
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تأليف
Yidong Sun
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A {em Motzkin path} of length $n$ is a lattice path from $(0,0)$ to $(n,0)$ in the plane integer lattice $mathbb{Z}timesmathbb{Z}$ consisting of horizontal-steps $(1, 0)$, up-steps $(1,1)$, and down-steps $(1,-1)$, which never passes below the x-axis. A {em $u$-segment {rm (resp.} $h$-segment {rm)}} of a Motzkin path is a maximum sequence of consecutive up-steps ({rm resp.} horizontal-steps). The present paper studies two kinds of statistics on Motzkin paths: number of $u$-segments and number of $h$-segments. The Lagrange inversion formula is utilized to represent the weighted generating function for the number of Motzkin paths according to the statistics as a sum of the partial Bell polynomials or the potential polynomials. As an application, a general framework for studying compositions are also provided.
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Two subclasses of Motzkin paths, S-Motzkin and T-Motzkin paths, are introduced. We provide bijections between S-Motzkin paths and ternary trees, S-Motzkin paths and non-crossing trees, and T-Motzkin paths and ordered pairs of ternary trees. Symbolic
equations for both paths, and thus generating functions for the paths, are provided. Using these, various parameters involving the two paths are analyzed.
A {em k-generalized Dyck path} of length $n$ is a lattice path from $(0,0)$ to $(n,0)$ in the plane integer lattice $mathbb{Z}timesmathbb{Z}$ consisting of horizontal-steps $(k, 0)$ for a given integer $kgeq 0$, up-steps $(1,1)$, and down-steps $(1,-
1)$, which never passes below the x-axis. The present paper studies three kinds of statistics on $k$-generalized Dyck paths: number of $u$-segments, number of internal $u$-segments and number of $(u,h)$-segments. The Lagrange inversion formula is used to represent the generating function for the number of $k$-generalized Dyck paths according to the statistics as a sum of the partial Bell polynomials or the potential polynomials. Many important special cases are considered leading to several surprising observations. Moreover, enumeration results related to $u$-segments and $(u,h)$-segments are also established, which produce many new combinatorial identities, and specially, two new expressions for Catalan numbers.
We introduce a new concept of permutation avoidance pattern called hatted pattern, which is a natural generalization of the barred pattern. We show the growth rate of the class of permutations avoiding a hatted pattern in comparison to barred pattern
. We prove that Dyck paths with no peak at height $p$, Dyck paths with no $ud... du$ and Motzkin paths are counted by hatted pattern avoiding permutations in $s_n(132)$ by showing explicit bijections. As a result, a new direct bijection between Motzkin paths and permutations in $s_n(132)$ without two consecutive adjacent numbers is given. These permutations are also represented on the Motzkin generating tree based on the Enumerative Combinatorial Object (ECO) method.
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