Catalan, Motzkin, and Riordan numbers

DOI10.1016/S0012-365X(99)00054-0zbMath0933.05002WikidataQ56442436 ScholiaQ56442436MaRDI QIDQ1300967

Frank R. Bernhart

Publication date: 2 September 1999

Published in: Discrete Mathematics (Search for Journal in Brave)

zbMATH Keywords

generating functionCatalan numbersMotzkin numberschromatic polynomialsLagrange inversionRiordan numbersdifference trianglesplanar coloring

Mathematics Subject Classification ID

Exact enumeration problems, generating functions (05A15) Combinatorial identities, bijective combinatorics (05A19) Enumeration in graph theory (05C30) Special sequences and polynomials (11B83)

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Cites Work

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• A correspondence between ordered trees and noncrossing partitions
• Hilbert series of fixed free algebras and noncommutative classical invariant theory
• The book thickness of a graph
• Rhyming schemes: crossings and coverings
• Automorphisms on Catalan trees and bracketings
• On the Birkhoff-Lewis equations
• On the structure of the lattice of noncrossing partitions
• Enumeration of plane trees by branches and endpoints
• A Catalan triangle
• A short proof of an identity of Touchard's concerning Catalan numbers
• Restricted plane tree representations of four Motzkin-Catalan equations
• Similarity relations on finite ordered sets
• Moments of Dyck paths
• Dyck path enumeration
• Enumeration of ladder graphs
• Positive Definite Matrices and Catalan Numbers, Revisited
• Counting Interval Graphs
• THE ENUMERATION OF A FAMILY OF LADDER GRAPHS PART II: SCHRODER AND SUPERCONNECTIVE RELATIONS
• THE ENUMERATION OF A FAMILY OF LADDER GRAPHS
• An Elementary Evaluation of the Catalan Numbers
• Embedding Graphs in Books: A Layout Problem with Applications to VLSI Design
• Triangulated polygons and frieze patterns
• Historical Note on a Recurrent Combinatorial Problem
• A note on plane trees
• An introduction to chromatic polynomials
• Coloring Six-Rings
• Relations between hypersurface cross ratios, and a combinatorial formula for partitions of a polygon, for permanent preponderance, and for non-associative products
• Six-Rings in Minimal Five-Color Maps
• A Contribution to the Theory of Chromatic Polynomials
• Chromatic Polynomials
• Motzkin numbers