Refinements of Lattice paths with flaws
| dc.creator | Ma, Jun | |
| dc.creator | Yeh, Yeong-Nan | |
| dc.date | 2008-12-15 | |
| dc.date.accessioned | 2026-07-07T12:12:55Z | |
| dc.date.available | 2026-07-07T12:12:55Z | |
| dc.description | The classical Chung-Feller theorem [2] tells us that the number of Dyck paths of length $n$ with $m$ flaws is the $n$-th Catalan number and independent on $m$. In this paper, we consider the refinements of Dyck paths with flaws by four parameters, namely peak, valley, double descent and double ascent. Let ${p}_{n,m,k}$ be the number of all the Dyck paths of semi-length $n$ with $m$ flaws and $k$ peaks. First, we derive the reciprocity theorem for the polynomial $P_{n,m}(x)=\sum\limits_{k=1}^np_{n,m,k}x^k$. Then we find the Chung-Feller properties for the sum of $p_{n,m,k}$ and $p_{n,m,n-k}$. Finally, we provide a Chung-Feller type theorem for Dyck paths of length $n$ with $k$ double ascents: the number of all the Dyck paths of semi-length $n$ with $m$ flaws and $k$ double ascents is equal to the number of all the Dyck paths that have semi-length $n$, $k$ double ascents and never pass below the x-axis, which is counted by the Narayana number. Let ${v}_{n,m,k}$ (resp. $d_{n,m,k}$) be the number of all the Dyck paths of semi-length $n$ with $m$ flaws and $k$ valleys (resp. double descents). Some similar results are derived. | |
| dc.identifier | https://arxiv.org/abs/0812.2820 | |
| dc.identifier | http://arxiv.org/abs/0812.2820 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/210702 | |
| dc.subject | Combinatorics | |
| dc.title | Refinements of Lattice paths with flaws | |
| dc.type | text |