Abstract
We consider the system of equations $A_k(x)=p(x)A_{k-1}(x)(q(x)+\sum_{i=0}^k A_i(x))$ for $k\geq r+1$ where $A_i(x)$, $0\leq i \leq r$, are some given functions and show how to obtain a close form for $A(x)=\sum_{k\geq 0}A_k(x)$. 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 $\pi$.
Highlights
Introduction and notationsLet A be a combinatorial class, that is a collection of similar objects endowed with a size function whose values are non-negative integers, so that the number | −1(n)| of objects of a given size n is finite
I=0 anchored with initial condition D0(x) = 1. They deduce algebraically a close form of the generating function D(x) = k 0 Dk(x) for Dh, by solving System (2), and prove that Dh, is enumerated by the Motzkin sequence, that is A001006 in the On-Line Encyclopedia of Integer Sequences (OEIS) [13]
Except Dyck and Motzkin cases, where we obtain two sequences appearing in [13] for the pattern U, all the others have never been studied in the literature
Summary
Let A be a combinatorial class, that is a collection of similar objects (lattices, trees, permutations, words) endowed with a size function whose values are non-negative integers, so that the number | −1(n)| of objects of a given size n is finite. H(U αD) h(β), where h is the height statistic, i.e. the maximal ordinate reached by a path In these studies, the authors need to consider a system of equations involving the generating functions Dk(x), k 0, for the subset of paths P ∈ Dh, satisfying h(P ) = k: k. The height of an occurrence of a pattern in a given path is the maximal ordinate reached by its points. We define the statistic hπ on lattice paths so that hπ(P ) is the maximal height reached by the occurrences of π in P.
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