Refinements of (n,m)(n,m)-Dyck paths

The classical Chung–Feller theorem tells us that the number of (n,m)(n,m)-Dyck paths is the nnth Catalan number and independent of mm. In this paper, we consider refinements of (n,m)(n,m)-Dyck paths by using four parameters, namely the peak, valley, double descent and double ascent. Let pn,m,kpn,m,k be the total number of (n,m)(n,m)-Dyck paths with kk peaks. First, we derive the reciprocity theorem for the polynomial Pn,m(x)=∑k=1npn,m,kxk. In particular, we prove that the number of (n,m)(n,m)-Dyck paths with kk peaks is equal to the number of (n,n−m)(n,n−m)-Dyck paths with n−kn−k peaks. Then we find the Chung–Feller properties for the sum of pn,m,kpn,m,k and pn,m,n−kpn,m,n−k, i.e., the number of (n,m)(n,m)-Dyck paths which have kk or n−kn−k peaks is 2(n+2)n(n−1)nk−1nk+1 for 1≤m≤n−11≤m≤n−1 and independent of mm. Finally, we provide a Chung–Feller type theorem for Dyck paths of semilength nn with kk double ascents: the total number of (n,m)(n,m)-Dyck paths with kk double ascents is equal to the total number of nn-Dyck paths that have kk double ascents and never pass below the xx-axis, which is counted by the Narayana number. Let vn,m,kvn,m,k (resp. dn,m,kdn,m,k) be the total number of (n,m)(n,m)-Dyck paths with kk valleys (resp. double descents). Some similar results are derived.