{"schema":"https://ver.cy/schemas/card/1.0.0","id":"vr.tr.motzkin-number","code":"thing-q2915234","url":"https://ver.cy/models/thing/q2915234/","name":"Motzkin number","alternateNames":[],"kind":"thing","status":"published","version":"0.2.0-wave.3","language":"en","classifiers":{"family":"Thing Registry","category":"Cross-cutting context","entryKind":"thing","plane":"XCT","domain":["XCT.QTY"],"industry":[],"navPath":"","tags":[],"facets":{}},"whatItIs":"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.","purpose":"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.","scope":{"in":[],"out":[],"boundaries":[]},"distinguishingFeatures":["Non-crossing chord interpretation","Motzkin path interpretation","Three-term recurrence","Algebraic generating function"],"structure":{"bundles":[{"id":"define","name":"Define","description":"What Motzkin numbers are.","layers":[{"id":"definition","name":"Definition","description":"Definition and interpretations.","findings":[{"id":"definition-finding","name":"Definition","description":"Definition.","questions":[{"text":"What are Motzkin numbers, and which combinatorial objects do they count?","kind":"definition"},{"text":"Is the question about Motzkin, Catalan or another related sequence?","kind":"boundary"}]}]},{"id":"relations","name":"Relations","description":"Relation to Catalan numbers.","findings":[{"id":"relations-finding","name":"Relations","description":"Relations.","questions":[{"text":"How are Motzkin numbers expressed in terms of Catalan numbers?","kind":"definition"},{"text":"Which entry fits Catalan number?","kind":"action"}]}]}]},{"id":"compute","name":"Compute","description":"Computation.","layers":[{"id":"recurrence","name":"Recurrence","description":"Recurrence and formulas.","findings":[{"id":"recurrence-finding","name":"Recurrence","description":"Recurrence.","questions":[{"text":"What recurrence, closed forms and generating function give Motzkin numbers?","kind":"action"},{"text":"What is the term for the index in question?","kind":"measurement"}]}]},{"id":"asymptotics","name":"Asymptotics","description":"Growth.","findings":[{"id":"asymptotics-finding","name":"Asymptotics","description":"Asymptotics.","questions":[{"text":"How fast do Motzkin numbers grow, and what is their asymptotic form?","kind":"provenance"},{"text":"Which references are standard?","kind":"provenance"}]}]}]},{"id":"theory","name":"Theory","description":"Properties and generalisations.","layers":[{"id":"properties","name":"Properties","description":"Properties.","findings":[{"id":"properties-finding","name":"Properties","description":"Properties.","questions":[{"text":"What number-theoretic and combinatorial properties do Motzkin numbers have?","kind":"provenance"},{"text":"Which entry fits enumerative combinatorics?","kind":"action"}]}]},{"id":"generalisations","name":"Generalisations","description":"Generalisations.","findings":[{"id":"generalisations-finding","name":"Generalisations","description":"Generalisations.","questions":[{"text":"What generalisations and refinements exist, such as Motzkin triangles and coloured paths?","kind":"provenance"},{"text":"Which sources are cited?","kind":"provenance"}]}]}]},{"id":"context","name":"Context","description":"History and applications.","layers":[{"id":"history","name":"History","description":"History.","findings":[{"id":"history-finding","name":"History","description":"History.","questions":[{"text":"How did Motzkin introduce the numbers in 1948, and how were interpretations discovered?","kind":"provenance"},{"text":"Which entry fits the history of combinatorics?","kind":"action"}]}]},{"id":"applications","name":"Applications","description":"Applications.","findings":[{"id":"applications-finding","name":"Applications","description":"Applications.","questions":[{"text":"Where do Motzkin numbers appear in computer science and physics?","kind":"provenance"},{"text":"Which entry fits lattice path?","kind":"action"}]}]}]}]},"agentConduct":{"may":["Compute Motzkin numbers and explain what they count.","Cite the sequence from recognised integer sequence references."],"mustNot":["Fabricate values, identities or proofs about the sequence.","Present conjectures as theorems.","Confuse Motzkin numbers with Catalan numbers or other sequences."],"requiresHuman":["Submitting proofs or results under a person's name."]},"ethics":{"considerations":["Mathematical credit belongs to those who did the work.","Errors in published sequences spread to every later use."],"affectedParties":["Students and researchers","Readers","Authors of results"]},"owners":{"steward":"Nobody owns a number sequence; it is held in common.","roles":[],"masterSystems":[]},"relations":[{"target":"Q16317911","type":"parent","note":"registry parent class"},{"target":"positive integer","type":"related","note":"in registry terms"},{"target":"integer sequence","type":"related","note":"in combinatorics"},{"target":"Catalan number","type":"related","note":"by summation formula"},{"target":"Theodore Motzkin","type":"related","note":"the mathematician"}],"interaction":{"identity":{"applicability":"required","items":["Vercy registry: vr.tr.motzkin-number","Wikidata: Q2915234 (https://www.wikidata.org/wiki/Q2915234)","OEIS: A001006 Motzkin numbers"]},"properties":{"applicability":"not-applicable","items":["first terms: 1, 1, 2, 4, 9, 21, 51, 127, 323, 835 sequence"]},"recognition":{"applicability":"optional","items":["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":{"applicability":"required","items":["define the sequence and interpretations","relay recurrence and generating function","relate to Catalan numbers","compute or verify terms"]},"hazards":{"applicability":"required","items":["Confusing with Catalan numbers","Indexing differences between sources","Computation errors for large terms"]},"interfaces":{"applicability":"required","items":["No regulation; the OEIS is the reference database"]},"context":{"applicability":"required","items":["Counting combinatorial structures.","Motzkin numbers","Motzkin paths and their variants","Motzkin triangle and refinements","related sequences such as Catalan, Schroder and Riordan numbers"]}},"sources":[{"title":"Wikidata item Q2915234: Motzkin number","url":"https://www.wikidata.org/wiki/Q2915234","note":"identity and sense of the item"},{"title":"Wikipedia: Motzkin number","url":"https://en.wikipedia.org/wiki/Motzkin_number","note":"general description of the item"}],"openQuestions":["Should Motzkin path be a separate entry?","How should the OEIS be linked?","How should combinatorial interpretations be catalogued?"],"resources":{"spec":"/models/things/publications/thing-q2915234/spec.json"},"provenance":{"origin":"thing registry research (pass 2)","builtFrom":["models/things/publications/thing-q2915234/spec.json"],"providers":["Claude"],"researchStatus":"unreviewed","generatedAt":"2026-09-12T23:27:52Z","builder":"tools/build_cards.py@1.0.0"},"completeness":{"sections":{"classifiers":"filled","whatItIs":"filled","purpose":"filled","distinguishingFeatures":"filled","structure":"filled","agentConduct":"filled","ethics":"filled","owners":"filled","relations":"filled","interaction.identity":"filled","interaction.properties":"not-applicable","interaction.recognition":"filled","interaction.capabilities":"filled","interaction.hazards":"filled","interaction.interfaces":"filled","interaction.context":"filled","sources":"filled"},"notes":{"_gate":"Published through the thing publication gate 1.0 on 2026-10-06T19:51:58+00:00: completeness 1.00; sense: Wikidata description; 2 live sources; review by codex: pass","interaction.properties":"Plane XCT: no invented physical properties."},"score":1.0}}