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

Abstract: 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.