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

For integers n,m with n≥1 and 0≤m≤n, an (n,m)-Dyck path is a lattice path in the integer lattice Z×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 Cn=1n+1(2nn). For any integer k with 1≤k≤n, let pn,m,k be the total number of (n,m)-Dyck paths with k peaks. Ma and Yeh proved that pn,m,k=pn,n−m,n−k for 0≤m≤n, and pn,m,k+pn,m,n−k=pn,m+1,k+pn,m+1,n−k for 1≤m≤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 pn,m,k and pn,m,k+pn,m,n−k according to the starting and ending steps.