|
| | | | |
| | | |
| Stat |
Members: 3645
Articles: 2'500'096
Articles rated: 2609
18 April 2024 |
|
| | | | |
|
Article overview
| |
|
|
Refinements of Lattice paths with flaws | | Jun Ma
; Yeong-Nan Yeh
; | | Date: |
15 Dec 2008 | | Abstract: | 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 $P_{n,m}(x)=sumlimits_{k=1}^np_{n,m,k}x^k$. 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. | | Source: | arXiv, 0812.2820 | | Services: | Forum | Review | PDF | Favorites | |
|
|
No review found.
Did you like this article?
Note: answers to reviews or questions about the article must be posted in the forum section.
Authors are not allowed to review their own article. They can use the forum section.
browser claudebot
|
| |
|
|
|
|
News, job offers and information for researchers and scientists:
| |
REGISTER