Parametric Catalan numbers and Catalan triangles. (English) Zbl 1257.05003
Summary: Here presented a generalization of Catalan numbers and Catalan triangles associated with two parameters based on the sequence characterization of Bell-type Riordan arrays. Among the generalized Catalan numbers, a class of large generalized Catalan numbers and a class of small generalized Catalan numbers are defined, which can be considered as an extension of large Schröder numbers and small Schröder numbers, respectively.
Using the characterization sequences of Bell-type Riordan arrays, some properties and expressions including the Taylor expansions of generalized Catalan numbers are given. A few combinatorial interpretations of the generalized Catalan numbers are also provided. Finally, a generalized Motzkin numbers and Motzkin triangles are defined similarly. An interrelationship among parametrical Catalan triangle, Pascal triangle, and Motzkin triangle is presented based on the sequence characterization of Bell-type Riordan arrays.
Using the characterization sequences of Bell-type Riordan arrays, some properties and expressions including the Taylor expansions of generalized Catalan numbers are given. A few combinatorial interpretations of the generalized Catalan numbers are also provided. Finally, a generalized Motzkin numbers and Motzkin triangles are defined similarly. An interrelationship among parametrical Catalan triangle, Pascal triangle, and Motzkin triangle is presented based on the sequence characterization of Bell-type Riordan arrays.
MSC:
| 05A05 | Permutations, words, matrices |
| 05A15 | Exact enumeration problems, generating functions |
| 15B36 | Matrices of integers |
| 05A30 | \(q\)-calculus and related topics |
| 05A10 | Factorials, binomial coefficients, combinatorial functions |
| 05A19 | Combinatorial identities, bijective combinatorics |
Keywords:
Catalan number; Catalan triangles; large generalized Catalan numbers; small generalized Catalan numbers; large Schröder numbers; small Schröder numbers; Motzkin numbers; Motzkin triuangles; Riordan array; Bell-type Riordan array; characteristic sequenceReferences:
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