Dyck Paths, Motzkin Paths, and the Binomial Transform.
Dyck Paths, Motzkin Paths, and the Binomial Transform.
Summary: We study the moments of orthogonal polynomial sequences (OPS) arising from tridiagonal matrices. We obtain combinatorial information about the sequence of moments of some OPS in terms of Motzkin and Dyck paths, and also in terms of the binomial transform. We then introduce an equivalence relation on the set of Dyck paths and some operations on them. We determine a formula for the cardinality of those equivalence classes, and use this information to obtain a combinatorial formula for the number of Dyck and Motzkin paths of a fixed length.
Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.), binomial transform, Motzkin path, Exact enumeration problems, generating functions, Dyck path, Motzkin path, binomial transform, Dyck path, Enumerative combinatorics
Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.), binomial transform, Motzkin path, Exact enumeration problems, generating functions, Dyck path, Motzkin path, binomial transform, Dyck path, Enumerative combinatorics
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