{ "model": { "rank": 5333, "code": "thing-q161519", "model_id": "vr.tr.permutation", "name": "permutation", "purpose": "Let an agent explain permutations and count them, describe their structure and special kinds, apply them in probability and computing, and distinguish them from combinations.", "family": "Thing Registry", "category": "Cross-cutting context", "status": "research-draft", "kind": "thing", "plane": "XCT", "domain": "XCT.QLT", "industry": "", "version": "", "url": "/models/thing/q161519/", "tier": 2, "score": 68, "payload": { "layer": "wikidata", "aliases": [ "transposition", "permutation involution", "pattern avoidance", "shader permutation", "derangement", "riffle shuffle permutation", "stack-sortable permutation", "cyclic permutation", "permutation of multiset", "random permutation", "Baxter permutation", "even permutation", "odd permutation", "circular permutation in proteins" ], "aliasCount": 14, "merged": 15, "knownIn": 68, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": null, "source": { "dataset": "wikidata", "item": "Q161519", "url": "https://www.wikidata.org/wiki/Q161519", "license": "CC0 1.0" } }, "research": { "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-11T22:38:06Z", "providers": [ "Claude" ], "breadth": "written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify", "pass": 2, "wave": 2, "engine": "claude" }, "metaModel": { "id": "THING-Q161519", "registryId": "vr.tr.permutation", "name": "permutation", "version": "0.2.0-wave.2", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.QLT" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q161519/", "model": { "registry_id": "vr.tr.permutation", "name": "permutation", "purpose": "Let an agent explain permutations and count them, describe their structure and special kinds, apply them in probability and computing, and distinguish them from combinations.", "definition": "An arrangement of the elements of a set in a particular order, or equivalently a bijection from a set to itself, with the number of permutations of n distinct items equal to n factorial, and with special kinds such as transpositions, involutions, derangements, cyclic permutations and riffle shuffle permutations; permutations are fundamental in combinatorics, group theory, probability, cryptography and computing.", "what_it_is_for": "Orderings and rearrangements.", "affordances": [ "explain and count permutations", "describe structure and special kinds", "apply in probability and computing", "distinguish from combinations" ], "distinguishing_features": [ "Order matters", "Bijection", "Group structure", "Counting principle" ], "appearance": "Not physical; orderings and mappings.", "visual_identification": [ "Ordered arrangement or bijection", "n factorial of n items", "A combination ignores order" ], "physical_properties": [], "families_and_kinds": [ "permutations of a finite set", "transpositions and cycles", "involutions and derangements", "pattern-avoiding permutations", "permutations in shuffling and computing" ], "related_models": [ { "relation": "is a kind of", "target": "bijection", "why": "category" }, { "relation": "is related to", "target": "sum", "why": "sums over permutations" }, { "relation": "is related to", "target": "homomorphism", "why": "permutation groups" }, { "relation": "is related to", "target": "ordered pair", "why": "ordered structures" } ], "identifiers": [], "standards_and_regulation": [ "Mathematical notation conventions" ], "failure_modes_and_hazards": [ "Confusing permutations with combinations", "Overflow in factorial computation", "Biased shuffling algorithms" ], "in_scope": [], "out_of_scope": [], "characteristics": [] }, "sources": [], "structure": { "bundles": [ { "id": "define", "name": "Define", "description": "What permutations are.", "rationale": "Concepts.", "layers": [ { "id": "definition", "name": "Definition", "description": "Definition and counting.", "findings": [ { "id": "definition-finding", "name": "Definition", "description": "Definition.", "questions": [ { "text": "What is a permutation, how many permutations of n items exist, and how does this differ from combinations?", "kind": "definition" }, { "text": "How are permutations written in one-line and cycle notation?", "kind": "definition" } ] } ] }, { "id": "count", "name": "Count", "description": "Counting problems.", "findings": [ { "id": "count-finding", "name": "Count", "description": "Counting.", "questions": [ { "text": "How many permutations satisfy this constraint, such as derangements or arrangements with repeats?", "kind": "action" }, { "text": "How are partial permutations counted?", "kind": "definition" } ] } ] } ] }, { "id": "structure", "name": "Structure", "description": "Structure of permutations.", "rationale": "Algebra.", "layers": [ { "id": "cycles", "name": "Cycles", "description": "Cycles and transpositions.", "findings": [ { "id": "cycles-finding", "name": "Cycles", "description": "Cycles.", "questions": [ { "text": "How are permutations decomposed into cycles and transpositions, and what are parity and sign?", "kind": "definition" }, { "text": "What are involutions?", "kind": "definition" } ] } ] }, { "id": "groups", "name": "Groups", "description": "Permutation groups.", "findings": [ { "id": "groups-finding", "name": "Groups", "description": "Groups.", "questions": [ { "text": "What is the symmetric group, and how do permutation groups arise in algebra?", "kind": "definition" }, { "text": "What is pattern avoidance in permutations?", "kind": "definition" } ] } ] } ] }, { "id": "apply", "name": "Apply", "description": "Applications.", "rationale": "Practice.", "layers": [ { "id": "probability", "name": "Probability", "description": "Probability and shuffling.", "findings": [ { "id": "probability-finding", "name": "Probability", "description": "Probability.", "questions": [ { "text": "How are permutations used in probability, and how many riffle shuffles randomise a deck, according to research?", "kind": "provenance" }, { "text": "What is a derangement probability?", "kind": "definition" } ] } ] }, { "id": "computing", "name": "Computing", "description": "Algorithms.", "findings": [ { "id": "computing-finding", "name": "Computing", "description": "Computing.", "questions": [ { "text": "How are permutations generated and shuffled correctly in code, and where do they appear in cryptography and graphics?", "kind": "action" }, { "text": "Which entry fits sorting or shader permutations?", "kind": "action" } ] } ] } ] }, { "id": "learn", "name": "Learn", "description": "Teaching and history.", "rationale": "Education.", "layers": [ { "id": "teach", "name": "Teach", "description": "Teaching permutations.", "findings": [ { "id": "teach-finding", "name": "Teach", "description": "Teaching.", "questions": [ { "text": "How can permutations be taught with hands-on counting?", "kind": "action" }, { "text": "Which misconceptions arise?", "kind": "provenance" } ] } ] }, { "id": "history", "name": "History", "description": "History.", "findings": [ { "id": "history-finding", "name": "History", "description": "History.", "questions": [ { "text": "How did the study of permutations develop, from early combinatorics to group theory?", "kind": "provenance" }, { "text": "Which references are standard?", "kind": "provenance" } ] } ] } ] } ] }, "openQuestions": [ "Should permutation groups be a separate entry?", "How should combinatorics resources be linked?", "How should algorithms be linked?" ], "statistics": { "bundles": 4, "layers": 8, "findings": 8, "questions": 16 } }, "draft": { "generator": "vr.draft.v3", "status": "draft-generated", "researched": false, "archetype": "abstract concept", "method": "Written from the archetype playbook - what this kind of thing needs beyond identity and provenance - and from the structure that recurred across 6,333 models already researched by two engines. Applied to this entry by rule. No source was read for this thing and no claim here is researched. This entry carries no facets of its own, so they were inferred from its domain - a guess about a whole domain applied to one thing.", "facetsInferred": true, "nextPass": "A researcher replaces this draft with a sourced specification. Treat every sentence below as a proposal to argue with.", "purpose": "Give an agent a durable, checkable way to recognise a permutation, record what state it is in, and decide what may be done with it.", "whatItIs": "Let an agent explain permutations and count them, describe their structure and special kinds, apply them in probability and computing, and distinguish them from combinations.", "characteristics": { "substance": "abstract", "origin": "conceptual", "agency": "inert" }, "whatYouCanDoWithIt": [ "observed and measured" ], "distinguishingFeatures": [ "Names folded into this entry, which a task may need to split apart again: transposition, permutation involution, pattern avoidance, shader permutation, derangement, riffle shuffle permutation, stack-sortable permutation, cyclic permutation, permutation of multiset, random permutation, Baxter permutation, even permutation.", "15 finer distinctions are held as aliases rather than separate entries, because telling them apart needs a task that asks for it.", "Described in 68 Wikipedia languages, which is a measure of how widely the thing is known, not of how important it is." ], "openQuestionsForResearch": [ "Which of the bundles below does a real task actually need, and which are ceremony?", "What does this thing have that the facets do not capture at all?", "Which neighbouring kind is most often confused with a permutation, and on what evidence are they told apart?" ], "whatItIsMadeOf": "an abstraction with no physical instance", "physicalCharacter": [ "Does nothing on its own; everything it does, something else did to it.", "These come from the domain this entry sits in rather than from the entry itself, so treat them as a first guess about the whole domain applied to one thing." ], "whatCanBeDoneWithIt": [ "observe it, measure it, record its state" ], "howItIsRecognised": [ "Nothing to see. What is recognised is an instance of it, and which instances count is exactly what is argued about." ], "relatedModels": [ { "relation": "covers", "note": "Finer kinds folded into this entry because telling them apart needs a task that asks for it. Each is a model waiting to be split out when one does.", "targets": [ "transposition", "permutation involution", "pattern avoidance", "shader permutation", "derangement", "riffle shuffle permutation", "stack-sortable permutation", "cyclic permutation", "permutation of multiset", "random permutation", "Baxter permutation", "even permutation" ] } ], "standing": "Described in 68 Wikipedia languages, which measures how widely it is written about rather than how important or how common it is. 15 finer distinctions are held inside this entry as names rather than as separate models.", "structure": { "bundles": [ { "id": "identity-and-classification", "name": "Identity, naming and classification", "description": "How an agent tells one permutation from another, and a permutation from things that resemble it.", "rationale": "Recognition comes before every other claim. Without stable identity nothing else in the model can be trusted to be about the same thing twice.", "layers": [ { "id": "naming-and-identifiers", "name": "Names and identifiers", "description": "The names this thing goes by and the identifiers that survive translation and time.", "findings": [ { "id": "preferred-name-and-aliases", "name": "Preferred name, aliases and local names", "description": "Which name to use, which names mean the same thing, and which merely sound similar.", "questions": [ { "id": "preferred-name-and-aliases-q01", "text": "What identifies and describes the name of a permutation, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "preferred-name-and-aliases-q02", "text": "Who or what asserted this about the name of a permutation, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "preferred-name-and-aliases-q03", "text": "What may an agent decide or do once the name of a permutation is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] }, { "id": "stable-identifiers", "name": "Stable identifiers and external keys", "description": "Identifiers that keep pointing at this kind of thing across systems and languages.", "questions": [ { "id": "stable-identifiers-q01", "text": "What identifies and describes an identifier for a permutation, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "stable-identifiers-q02", "text": "Who or what asserted this about an identifier for a permutation, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "stable-identifiers-q03", "text": "What may an agent decide or do once an identifier for a permutation is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "classification-and-granularity", "name": "Classification and granularity", "description": "Where a permutation sits among kinds, and how finely a task needs to cut it.", "findings": [ { "id": "kind-and-parents", "name": "Kind, parents and neighbouring kinds", "description": "The classes this thing belongs to and the ones it is next to.", "questions": [ { "id": "kind-and-parents-q01", "text": "What identifies and describes the kind of a permutation, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "kind-and-parents-q02", "text": "Who or what asserted this about the kind of a permutation, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "kind-and-parents-q03", "text": "What may an agent decide or do once the kind of a permutation is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] }, { "id": "distinguishing-features", "name": "Distinguishing features", "description": "What separates a permutation from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a permutation, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "distinguishing-features-q02", "text": "Who or what asserted this about what distinguishes a permutation, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "distinguishing-features-q03", "text": "What may an agent decide or do once what distinguishes a permutation is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] }, { "id": "state-and-lifecycle", "name": "State and lifecycle", "description": "The states a permutation passes through and the events that move it between them.", "rationale": "Most decisions about a thing depend on what state it is in now, which is a claim with a time on it, not a property.", "layers": [ { "id": "lifecycle-stages", "name": "Lifecycle stages", "description": "From coming into existence to ceasing to be one of these.", "findings": [ { "id": "stages-and-transitions", "name": "Stages and transitions", "description": "The stages worth naming and what moves a permutation between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a permutation, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "stages-and-transitions-q02", "text": "Who or what asserted this about the lifecycle of a permutation, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "stages-and-transitions-q03", "text": "What may an agent decide or do once the lifecycle of a permutation is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "observations-and-status", "name": "Observations and current status", "description": "What is observed about a permutation, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a permutation is recorded so that it can be superseded rather than overwritten.", "questions": [ { "id": "observation-record-q01", "text": "What identifies and describes an observation of a permutation, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "observation-record-q02", "text": "Who or what asserted this about an observation of a permutation, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "observation-record-q03", "text": "What may an agent decide or do once an observation of a permutation is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] }, { "id": "definitions-in-contest", "name": "Definitions and who holds them", "description": "What permutation is taken to mean, and by whom.", "rationale": "When a field disagrees about a concept, the disagreement is the content. A model that picks one definition silently destroys the information.", "layers": [ { "id": "competing-definitions", "name": "Competing definitions", "description": "The main readings and the traditions behind them.", "findings": [ { "id": "definition-map", "name": "Definitions and their holders", "description": "Each definition with the school or body that holds it.", "questions": [ { "id": "definition-map-q01", "text": "Which definitions of permutation are in use, and which tradition or body holds each?", "kind": "definition" }, { "id": "definition-map-q02", "text": "What turns on the difference between them in practice?", "kind": "boundary" } ] } ] }, { "id": "operationalisation", "name": "Operationalisation", "description": "How it is measured or applied when it has to be.", "findings": [ { "id": "operational-record", "name": "Measures and proxies", "description": "Instruments and indicators used to stand in for it.", "questions": [ { "id": "operational-record-q01", "text": "How is permutation operationalised or measured in practice, and by what instrument?", "kind": "measurement" }, { "id": "operational-record-q02", "text": "What does that operationalisation leave out, and when does that matter?", "kind": "boundary" } ] } ] } ] }, { "id": "instances-and-use", "name": "Instances, use and consequence", "description": "What counts as an instance of permutation and what follows from calling something that.", "rationale": "Applying a concept is an act with consequences, so a model must say what the label licenses and what it does not.", "layers": [ { "id": "instances", "name": "What counts as an instance", "description": "Clear cases, borderline cases and non-cases.", "findings": [ { "id": "instance-tests", "name": "Tests for an instance", "description": "What would settle whether something falls under it.", "questions": [ { "id": "instance-tests-q01", "text": "What would settle whether something is an instance of permutation?", "kind": "boundary" }, { "id": "instance-tests-q02", "text": "Which borderline cases are argued about, and on what grounds?", "kind": "definition" } ] } ] }, { "id": "consequence", "name": "Consequence of application", "description": "Rights, duties or decisions that follow from the label.", "findings": [ { "id": "consequence-record", "name": "What the label licenses", "description": "What an agent may do once something is classified this way.", "questions": [ { "id": "consequence-record-q01", "text": "What follows practically once something is treated as permutation?", "kind": "action" }, { "id": "consequence-record-q02", "text": "What must an agent not infer from the label alone?", "kind": "action" } ] } ] } ] }, { "id": "provenance-and-evidence", "name": "Provenance, evidence and time", "description": "Where every claim about a permutation came from and when it held.", "rationale": "A claim without a source and a time cannot be superseded, only overwritten, and an agent that overwrites loses the ability to explain itself.", "layers": [ { "id": "source-and-authority", "name": "Source and authority", "description": "Who said it, on what evidence, and how strongly.", "findings": [ { "id": "claim-provenance", "name": "Claim provenance and confidence", "description": "The authority behind each claim about a permutation and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a permutation, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "claim-provenance-q02", "text": "Who or what asserted this about a claim about a permutation, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "claim-provenance-q03", "text": "What may an agent decide or do once a claim about a permutation is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "time-and-versions", "name": "Time, versions and supersession", "description": "When a claim was true, when it was learnt, and what replaced it.", "findings": [ { "id": "validity-and-supersession", "name": "Validity period and supersession", "description": "How an old claim about a permutation is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a permutation, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "validity-and-supersession-q02", "text": "Who or what asserted this about the validity of a claim about a permutation, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "validity-and-supersession-q03", "text": "What may an agent decide or do once the validity of a claim about a permutation is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] } ] }, "statistics": { "bundles": 5, "layers": 10, "findings": 12, "questions": 32 } } } }