Refinements of Lattice paths with flaws

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

Refinements of two identities on (n,m)(n,m)-Dyck paths

For integers $n, m$ with $n \geq 1$ and $0 \leq m \leq n$, an $(n,m)$-Dyck path is a lattice path in the integer lattice $\mathbb{Z} \times \mathbb{Z}$ using up steps $(0,1)$ and down steps $(1,0)$ that goes from the origin $(0,0)$ to the point $(n,n)$ and contains exactly $m$ up steps below the line $y=x$. The classical Chung-Feller theorem says that the total number of $(n,m)$-Dyck path is independent of $m$ and is equal to the $n$-th Catalan number $C_n=\frac{1}{n+1}{2n \choose n}$. For any integer $k$ with $1 \leq k \leq n$, let $p_{n,m,k}$ be the total number of $(n,m)$-Dyck paths with $k$ peaks. Ma and Yeh proved that $p_{n,m,k}$=$p_{n,n-m,n-k}$ for $0 \leq m \leq n$, and $p_{n,m,k}+p_{n,m,n-k}=p_{n,m+1,k}+p_{n,m+1,n-k}$ for $1 \leq m \leq n-2$. In this paper we give bijective proofs of these two results. Using our bijections, we also get refined enumeration results on the numbers $p_{n,m,k}$ and $p_{n,m,k}+p_{n,m,n-k}$ according to the starting and ending steps.Comment: 9 pages, with 2 figure

Chung-Feller property in view of generating functions

The classical Chung-Feller Theorem offers an elegant perspective for enumerating the Catalan number cn = 1 2n) n+1 n. One of the various proofs is by the uniformpartition method. The method shows that the set of the free Dyck n-paths, which have ( 2n) n in total, is uniformly partitioned into n + 1 blocks, and the ordinary Dyck n-paths form one of these blocks; therefore the cardinality of each block is 1 2n) n+1 n. In this article, we study the Chung-Feller property: a sup-structure set can be uniformly partitioned such that one of the partition blocks is (isomorphic to) a well-known structure set. The previous works about the uniform-partition method used bijections, but here we apply generating functions as a new approach. By claiming a functional equation involving the generating functions of sup- and sub-structure sets, we re-prove two known results about Chung-Feller property, and explore several new examples including the ones for the large and the little Schröder paths. Especially for the Schröder paths, we are led by the new approach straightforwardly to consider “weighted ” free Schröder paths as sup-structures. The weighted structures are not obvious via bijections or other methods. Partially supported by NSC 98-2115-M-134-005-MY3 Partially supported by NSFC 11071030 Partially supported by NSC 98-2115-M-001-019-MY