Parametric Catalan numbers and Catalan triangles

DOI10.1016/j.laa.2012.10.001zbMath1257.05003OpenAlexW2017381362MaRDI QIDQ1931755

Yanyan Li

Publication date: 16 January 2013

Published in: Linear Algebra and its Applications (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/j.laa.2012.10.001

zbMATH Keywords

Catalan numberMotzkin numbersRiordan arraycharacteristic sequencelarge Schröder numbersBell-type Riordan arrayCatalan triangleslarge generalized Catalan numbersMotzkin triuanglessmall generalized Catalan numberssmall Schröder numbers

Mathematics Subject Classification ID

Exact enumeration problems, generating functions (05A15) Factorials, binomial coefficients, combinatorial functions (05A10) Combinatorial identities, bijective combinatorics (05A19) (q)-calculus and related topics (05A30) Permutations, words, matrices (05A05) Matrices of integers (15B36)

Related Items (34)

Combinatorics of a generalized Narayana identity ⋮ Arithmetic into geometric progressions through Riordan arrays ⋮ Unnamed Item ⋮ Fuss-Catalan matrices, their weighted sums, and stabilizer subgroups of the Riordan group ⋮ Unnamed Item ⋮ Half of a Riordan array and restricted lattice paths ⋮ Positivity of Iterated Sequences of Polynomials ⋮ Explicit formulas and recurrence relations for generalized Catalan numbers ⋮ A generalization of Catalan numbers ⋮ On an extension of Riordan array and its application in the construction of convolution-type and Abel-type identities ⋮ The \(\mathfrak{uvu}\)-avoiding \((a, b, c)\)-generalized Motzkin paths with vertical steps: bijections and statistic enumerations ⋮ Half Riordan array sequences ⋮ Riordan arrays and difference equations of subdiagonal lattice paths ⋮ Schröder matrix as inverse of Delannoy matrix ⋮ Matrix characterizations of Riordan arrays ⋮ The Vertical Recursive Relation of Riordan Arrays and Their Matrix Representation ⋮ Generalized Schröder matrices arising from enumeration of lattice paths ⋮ Total positivity of Catalan triangle ⋮ Sequence characterizations of double Riordan arrays and their compressions ⋮ Row polynomial matrices of Riordan arrays ⋮ Some inverse relations determined by Catalan matrices ⋮ Determinant representations of polynomial sequences of Riordan type ⋮ Total positivity of Riordan arrays ⋮ Total positivity of recursive matrices ⋮ Palindromic Riordan arrays, classical orthogonal polynomials and Catalan triangles ⋮ A unified approach for the Catalan matrices by using Riordan arrays ⋮ \(A\)-sequences, \(Z\)-sequence, and \(B\)-sequences of Riordan matrices ⋮ Shift operators defined in the Riordan group and their applications ⋮ Log-convexity of Aigner-Catalan-Riordan numbers ⋮ Taylor expansions for the generating function of Catalan-like numbers ⋮ Palindromes and pseudo-involution multiplication ⋮ Centralizers of the Riordan group ⋮ Unnamed Item ⋮ Riordan array approach to Lagrange inversion formulas and Chung-Feller theory

Cites Work

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• Combinatorics of Riordan arrays with identical \(A\) and \(Z\) sequences
• Riordan arrays associated with Laurent series and generalized Sheffer-type groups
• An algebraic structure for Faber polynomials
• Sequence characterization of Riordan arrays
• The Riordan group
• The number of Baxter permutations
• Pascal triangles, Catalan numbers and renewal arrays
• Riordan arrays and combinatorial sums
• A classic proof of a recurrence for a very classical sequence
• Enumerating a class of lattice paths
• Permutations, parenthesis words, and Schröder numbers
• Taylor expansions for Catalan and Motzkin numbers
• Bijections and the Riordan group
• Riordan arrays and the Abel-Gould identity
• Generating trees and the Catalan and Schröder numbers
• A divisibility property for a subgroup of Riordan matrices
• Generating trees and proper Riordan arrays
• The Sheffer group and the Riordan group
• Riordan group involutions
• Hipparchus, Plutarch, Schroder, and Hough
• On Some Alternative Characterizations of Riordan Arrays
• ECO:a methodology for the enumeration of combinatorial objects
• Some open questions about random walks, involutions, limiting distributions, and generating functions

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