Motzkin number
Let an agent define Motzkin numbers, relay their combinatorial interpretations, recurrence and generating function, relate them to Catalan numbers, and help compute or verify terms.
Bundle → Layer → Finding → Questions Filled
4 bundles · 8 layers · 8 findings · 16 questions
Define What Motzkin numbers are.
Definition
Definition and interpretations.
Definition
Definition.
- What are Motzkin numbers, and which combinatorial objects do they count? definition
- Is the question about Motzkin, Catalan or another related sequence? boundary
Relations
Relation to Catalan numbers.
Relations
Relations.
- How are Motzkin numbers expressed in terms of Catalan numbers? definition
- Which entry fits Catalan number? action
Compute Computation.
Recurrence
Recurrence and formulas.
Recurrence
Recurrence.
- What recurrence, closed forms and generating function give Motzkin numbers? action
- What is the term for the index in question? measurement
Asymptotics
Growth.
Asymptotics
Asymptotics.
- How fast do Motzkin numbers grow, and what is their asymptotic form? provenance
- Which references are standard? provenance
Theory Properties and generalisations.
Properties
Properties.
Properties
Properties.
- What number-theoretic and combinatorial properties do Motzkin numbers have? provenance
- Which entry fits enumerative combinatorics? action
Generalisations
Generalisations.
Generalisations
Generalisations.
- What generalisations and refinements exist, such as Motzkin triangles and coloured paths? provenance
- Which sources are cited? provenance
Context History and applications.
History
History.
History
History.
- How did Motzkin introduce the numbers in 1948, and how were interpretations discovered? provenance
- Which entry fits the history of combinatorics? action
Applications
Applications.
Applications
Applications.
- Where do Motzkin numbers appear in computer science and physics? provenance
- Which entry fits lattice path? action
Classifiers Filled
- Family
- Thing Registry
- Category
- Cross-cutting context
- Entry kind
- thing
- Plane
- XCT
- Domain
- XCT.QTY
What it is Filled
A number in the sequence 1, 1, 2, 4, 9, 21, 51, 127, 323, 835 and so on, counting the ways to draw non-crossing chords between some of n points on a circle, and equivalently counting Motzkin paths and other combinatorial objects; Motzkin numbers are named after Theodore Motzkin, satisfy a recurrence related to the Catalan numbers, and appear in enumerative combinatorics and computer science.
Why it exists Filled
Let an agent define Motzkin numbers, relay their combinatorial interpretations, recurrence and generating function, relate them to Catalan numbers, and help compute or verify terms.
Distinguishing features Filled
- Non-crossing chord interpretation
- Motzkin path interpretation
- Three-term recurrence
- Algebraic generating function
What robots and AI may and may not do Filled
Must not
- Fabricate values, identities or proofs about the sequence.
- Present conjectures as theorems.
- Confuse Motzkin numbers with Catalan numbers or other sequences.
Only with a human decision
- Submitting proofs or results under a person's name.
May
- Compute Motzkin numbers and explain what they count.
- Cite the sequence from recognised integer sequence references.
Moral aspects Filled
- Mathematical credit belongs to those who did the work.
- Errors in published sequences spread to every later use.
Who is affected
- Students and researchers
- Readers
- Authors of results
Owners Filled
Steward
Nobody owns a number sequence; it is held in common.
Links to other meta-models Filled
parent
- Q16317911 - registry parent class
related
- positive integer - in registry terms
- integer sequence - in combinatorics
- Catalan number - by summation formula
- Theodore Motzkin - the mathematician
What else AI and robots need to interact with it Filled
Identity and identifiers required Filled
- Vercy registry: vr.tr.motzkin-number
- Wikidata: Q2915234 (https://www.wikidata.org/wiki/Q2915234)
- OEIS: A001006 Motzkin numbers
Direct properties not applicable Not applicable
- first terms: 1, 1, 2, 4, 9, 21, 51, 127, 323, 835 sequence
Plane XCT: no invented physical properties.
Recognition optional Filled
- Sequence 1, 1, 2, 4, 9, 21, 51
- Counts non-crossing chords and Motzkin paths
- Catalan numbers count full non-crossing matchings; Schroder numbers are another related sequence
- Not a visible object; an integer sequence and lattice path diagrams.
Capabilities and actions required Filled
- define the sequence and interpretations
- relay recurrence and generating function
- relate to Catalan numbers
- compute or verify terms
Hazards and failure modes required Filled
- Confusing with Catalan numbers
- Indexing differences between sources
- Computation errors for large terms
Standards and interfaces required Filled
- No regulation; the OEIS is the reference database
Context of use required Filled
- Counting combinatorial structures.
- Motzkin numbers
- Motzkin paths and their variants
- Motzkin triangle and refinements
- related sequences such as Catalan, Schroder and Riordan numbers
Sources Filled
- Wikidata item Q2915234: Motzkin number - identity and sense of the item
- Wikipedia: Motzkin number - general description of the item
Open questions
- Should Motzkin path be a separate entry?
- How should the OEIS be linked?
- How should combinatorial interpretations be catalogued?
Machine files
Provenance
thing registry research (pass 2) · unreviewed
Built from: models/things/publications/thing-q2915234/spec.json