Lattice paths with a first return decomposition constrained by the maximal height of a pattern
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We consider the system of equations A_k(x)=p(x)A_k-1(x)(q(x)+∑_i=0^k A_i(x)) for k≥ r+1 where A_i(x), 0≤ i ≤ r, are some given functions and show how to obtain a close form for A(x)=∑_k≥ 0A_k(x). We apply this general result to the enumeration of certain subsets of Dyck, Motzkin, skew Dyck, and skew Motzkin paths, defined recursively according to the first return decomposition with a monotonically non-increasing condition relative to the maximal ordinate reached by an occurrence of a given pattern π.
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