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 C_n=1/（n+1）（?）.For any integer k with 1≤k≤n,let p_n,m,kbe 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-kfor 0≤m≤n,and p_n,m,k+p_n,m,n-k=p_n,m+1,k+p_n,m+1,n-kfor 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 p_n,m,kand p_n,m,k+p_n,m,n-kaccording to the starting and ending steps.