Potential Polynomials and Motzkin Paths
| dc.creator | Sun, Yidong | |
| dc.date | 2008-05-28 | |
| dc.date.accessioned | 2026-07-07T09:41:22Z | |
| dc.date.available | 2026-07-07T09:41:22Z | |
| dc.description | A {\em Motzkin path} of length $n$ is a lattice path from $(0,0)$ to $(n,0)$ in the plane integer lattice $\mathbb{Z}\times\mathbb{Z}$ 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. | |
| dc.description | 11 pages, 1 figures; Discreste Math., to appear | |
| dc.identifier | https://arxiv.org/abs/0805.4358 | |
| dc.identifier | http://arxiv.org/abs/0805.4358 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/161802 | |
| dc.subject | Combinatorics | |
| dc.subject | 05A05; 05A15 | |
| dc.title | Potential Polynomials and Motzkin Paths | |
| dc.type | text |