Refinements of -Dyck paths

The classical Chung–Feller theorem tells us that the number of ( n , m ) -Dyck paths is the n th Catalan number and independent of m . In this paper, we consider refinements of ( n , m ) -Dyck paths by using four parameters, namely the peak, valley, double descent and double ascent. Let p n , m , k be the total number of ( n , m ) -Dyck paths with k peaks. First, we derive the reciprocity theorem for the polynomial P n , m ( x ) = ∑ k = 1 n p n , m , k x k . In particular, we prove that the number of ( n , m ) -Dyck paths with k peaks is equal to the number of ( n , n − m ) -Dyck paths with n − k peaks. Then we find the Chung–Feller properties for the sum of p n , m , k and p n , m , n − k , i.e., the number of ( n , m ) -Dyck paths which have k or n − k peaks is 2 ( n + 2 ) n ( n − 1 ) n k − 1 n k + 1 for 1 ≤ m ≤ n − 1 and independent of m . Finally, we provide a Chung–Feller type theorem for Dyck paths of semilength n with k double ascents: the total number of ( n , m ) -Dyck paths with k double ascents is equal to the total number of n -Dyck paths that have 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 total number of ( n , m ) -Dyck paths with k valleys (resp. double descents). Some similar results are derived.