2016/03/04 by Kangro, Kairi, Pourmoradnasseri, Mozhgan, Theis, Dirk Oliver
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1603.01422
A dispersed Dyck path (DDP) of length n is a lattice path on N× N from (0,0) to (n,0) in which the following steps are allowed: "up" (x, y) → (x+1, y+1); "down" (x, y) → (x+1, y-1); and "right" (x,0) → (x+1,0). An ascent in a DDP is an inclusion-wise maximal sequence of consecutive up steps. A 1-ascent is an ascent consisting of exactly 1 up step. We give a closed formula for the total number of 1-ascents in all dispersed Dyck paths of length n, A191386 in Sloane's OEIS. Previously, only implicit generating function relations and asymptotics were known.