Dyck paths with a first return decomposition constrained by height

Grand Dyck paths with air pockets

“…The o.g.f. that counts the partial GDAP above the line y " m (with respect to the length) is given by f p1q `gp1q " r ´m 2 ´r´1´m 2 ´x2 x 3 .…”

Grand Dyck paths with air pockets

“…where P , G 1 , G 2 , and G `are the generating functions with respect to the length for the cardinalities of P, G 1 , G 2 , and G `, respectively. We have P pxq " xApxq, where A is the generating function for the set A of Dyck paths with air pockets, see (1). Solving the system, we get:…”

“…Recently in [1], the authors introduced and enumerated the subset D h,ě of restricted Dyck paths defined as follows: the set D h,ě is the union of the empty Dyck path with all Dyck paths P having a first return decomposition P " U αDβ satisfying the conditions: " α, β P D h,ě , hpU αDq ě hpβq,…”

“…The next section recalls some useful results from [2] and introduces several notations for particular subsets of GDAP. The main result of Section 3 is Theorem 1 which gives the generating function counting the GDAP with respect to the length, which is the 'grand' counterpart of the generating function in (1) for DAP. In Section 4 we give similar results for prefixes of GDAP ending at a given ordinate, the corresponding problem for DAP being already solved in [10].…”

Pattern distributions in Dyck paths with a first return decomposition constrained by height

“…In three recent articles [1,2,6], the authors enumerate particular subclasses of lattice paths (Dyck, Motzkin, 2-Motzkin and Schröder paths) defined recursively according to the first return decomposition with a condition on the path height. More precisely, in the first paper, they focus on the Dyck path set D h, consisting of the union of the empty path with all Dyck paths P having a first return decomposition P = U αDβ satisfying the conditions:…”