{"schema":"https://ver.cy/schemas/card/1.0.0","id":"vr.tr.eulerian-number","code":"thing-q1373849","url":"https://ver.cy/models/thing/q1373849/","name":"Eulerian number","alternateNames":[],"kind":"thing","status":"published","version":"0.1.0-research.1","language":"en","classifiers":{"family":"Thing Registry","category":"Cross-cutting context","entryKind":"thing","plane":"XCT","domain":["XCT.QTY"],"industry":[],"navPath":"","tags":[],"facets":{}},"whatItIs":"An Eulerian number of the first order, A(n,k), counts permutations of n distinct elements having exactly k descents, where a descent is an adjacent pair whose first element exceeds its second.","purpose":"Enable an agent to identify, compute, validate and apply Eulerian numbers without confusing their indexing conventions or counted permutation statistics.","scope":{"in":["The permutation statistic defining an Eulerian number","Index meanings, notation translations and boundary conventions","Exact values, computation methods and verification evidence","Relations to Eulerian polynomials and permutation probabilities","Recognition and routing of generalized Eulerian families"],"out":["Euler's constant e and the Euler-Mascheroni constant","Euler numbers and Euler polynomials as separate mathematical families","General permutation representations and manipulation","Stirling numbers, binomial coefficients and their independent models","Physical quantities, measurement units and unit conversions"],"boundaries":[]},"distinguishingFeatures":["Under the adopted convention, A(n,k) counts permutations of n distinct ordered elements with exactly k adjacent descents. [NIST DLMF](https://dlmf.nist.gov/26.14#i)","Its identity requires both indices and their convention; an isolated integer or the name Euler is insufficient.","The zero-based row for n=3 is [1,4,1], providing a concrete indexing check. [NIST DLMF table](https://dlmf.nist.gov/26.14.T1)","A proposed interpretation must count adjacent descents rather than all inversions, descent-position sums or permutation cycles.","Treat signed, multiset or higher-order variants as requiring an explicit family definition before accepting their values."],"structure":{"bundles":[{"id":"counted-permutation-class","name":"Counted permutation class","description":"Establish what the number counts before interpreting any value.","layers":[{"id":"ordinary-family-definition","name":"Ordinary family definition","description":"Anchor the entry to permutations of distinct ordered elements.","findings":[{"id":"descent-count-anchor","name":"Descent count anchor","description":"Adopt the ordinary descent interpretation as the draft's working definition and record evidence for each imported interpretation.","questions":[{"text":"Does the source count permutations of n distinct ordered elements with exactly k positions i satisfying sigma(i) > sigma(i+1)?","id":"descent-count-anchor-q01","kind":"definition"},{"text":"Which read definition establishes that the supplied notation denotes this count?","id":"descent-count-anchor-q02","kind":"provenance"}]}]},{"id":"statistic-and-family-boundaries","name":"Statistic and family boundaries","description":"Resolve alternative statistics and proposed generalizations.","findings":[{"id":"interpretation-compatibility","name":"Interpretation compatibility","description":"Require an explicit translation or separate family declaration before reusing an imported count.","questions":[{"text":"If the source uses ascents, excedances or runs, what argument and index translation connect its count to the adopted convention?","id":"interpretation-compatibility-q01","kind":"boundary"},{"text":"Does the source introduce repeated elements, signs or an order parameter that changes the counted family?","id":"interpretation-compatibility-q02","kind":"boundary"}]}]}]},{"id":"indices-and-boundaries","name":"Indices and boundaries","description":"Make each Eulerian triangle position unambiguous.","layers":[{"id":"notation-translation","name":"Notation translation","description":"Preserve source indices alongside their canonical interpretation.","findings":[{"id":"column-meaning","name":"Column meaning","description":"Record whether a column labels the statistic directly or uses an offset.","questions":[{"text":"What do the source's first and second indices mean, and where does column numbering begin?","id":"column-meaning-q01","kind":"definition"},{"text":"Does translating the source's n=3 row recover canonical columns k=0,1,2 with values 1,4,1?","id":"column-meaning-q02","kind":"measurement"}]}]},{"id":"support-and-empty-case","name":"Support and empty case","description":"Separate combinatorial support from input validation.","findings":[{"id":"boundary-contract","name":"Boundary contract","description":"Adopt A(0,0)=1; for n>=1, support is 0<=k<=n-1. Extend other integer k values by zero; reject negative or noninteger n. [NIST DLMF](https://dlmf.nist.gov/26.14)","questions":[{"text":"Does the implementation preserve the empty-permutation value separately from positive-size rows?","id":"boundary-contract-q01","kind":"boundary"},{"text":"Will an out-of-support integer k return zero while malformed or unsupported indices produce an explicit error?","id":"boundary-contract-q02","kind":"action"}]}]}]},{"id":"exact-evaluation-and-checks","name":"Exact evaluation and checks","description":"Compute Eulerian counts and assess supplied representations.","layers":[{"id":"recurrence-evaluation","name":"Recurrence evaluation","description":"Connect each computed entry to initialized preceding rows.","findings":[{"id":"recurrence-contract","name":"Recurrence contract","description":"For n>=2, A(n,k)=(k+1)A(n-1,k)+(n-k)A(n-1,k-1). [NIST DLMF recurrence](https://dlmf.nist.gov/26.14.E8)","questions":[{"text":"Which base rows and zero-extension rules initialize the recurrence?","id":"recurrence-contract-q01","kind":"definition"},{"text":"Can the selected arithmetic represent every intermediate count exactly, or must the result be labelled approximate or modular?","id":"recurrence-contract-q02","kind":"action"}]}]},{"id":"independent-validation","name":"Independent validation","description":"Check individual values and complete rows using different evidence.","findings":[{"id":"row-invariants","name":"Row invariants","description":"For n>=1, require A(n,k)=A(n,n-1-k) and sum over k=0 through n-1 equal to n!. [NIST DLMF identities](https://dlmf.nist.gov/26.14#iii)","questions":[{"text":"Does the complete computed row satisfy symmetry and the factorial total?","id":"row-invariants-q01","kind":"measurement"},{"text":"Which small-case enumeration or independently evaluated formula checks the claimed cell beyond these necessary row tests?","id":"row-invariants-q02","kind":"provenance"}]}]}]},{"id":"polynomial-and-probability-use","name":"Polynomial and probability use","description":"Control how Eulerian counts enter algebraic and probabilistic tasks.","layers":[{"id":"polynomial-coefficients","name":"Polynomial coefficients","description":"Link counts to a declared polynomial representation.","findings":[{"id":"exponent-convention","name":"Exponent convention","description":"For this model's interface, define P_n(t)=sum_k A(n,k)t^k and require explicit translation from other normalizations.","questions":[{"text":"Does the referenced polynomial attach A(n,k) to t^k, t^(k+1) or another expression?","id":"exponent-convention-q01","kind":"definition"},{"text":"What exponent shift or normalization must the agent apply before extracting or exporting coefficients?","id":"exponent-convention-q02","kind":"action"}]}]},{"id":"descent-probabilities","name":"Descent probabilities","description":"Record the sampling assumption needed to normalize a count.","findings":[{"id":"uniform-sampling-condition","name":"Uniform sampling condition","description":"For a uniformly sampled permutation, the definition implies Pr(k descents)=A(n,k)/n!; record this as a derived probability.","questions":[{"text":"Are all n! permutations equally likely, or does the task impose weights or restrictions?","id":"uniform-sampling-condition-q01","kind":"boundary"},{"text":"Should the derived probability remain an exact rational or be rounded with a stated precision?","id":"uniform-sampling-condition-q02","kind":"action"}]}]}]},{"id":"evidence-and-external-alignment","name":"Evidence and external alignment","description":"What the world already says about this thing, gathered so the model can be checked against it.","layers":[{"id":"reported-evidence","name":"Reported evidence","description":"Findings from the breadth pass, kept separate from the structural claims.","findings":[{"id":"evidence-confidence-notes","name":"Check these first","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these check these first hold for the sense of Eulerian number this model covers, and on what evidence?","id":"evidence-confidence-notes-q01","kind":"provenance"}]},{"id":"evidence-kinds","name":"Kinds and varieties","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these kinds and varieties hold for the sense of Eulerian number this model covers, and on what evidence?","id":"evidence-kinds-q01","kind":"provenance"}]},{"id":"evidence-identifiers","name":"Identifiers and schemes","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these identifiers and schemes hold for the sense of Eulerian number this model covers, and on what evidence?","id":"evidence-identifiers-q01","kind":"provenance"}]},{"id":"evidence-real-world-use","name":"Real-world use","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these real-world use hold for the sense of Eulerian number this model covers, and on what evidence?","id":"evidence-real-world-use-q01","kind":"provenance"}]},{"id":"evidence-failure-modes-and-hazards","name":"Failure modes and hazards","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these failure modes and hazards hold for the sense of Eulerian number this model covers, and on what evidence?","id":"evidence-failure-modes-and-hazards-q01","kind":"provenance"}]},{"id":"evidence-neighbours","name":"Neighbouring kinds and how to tell them apart","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these neighbouring kinds and how to tell them apart hold for the sense of Eulerian number this model covers, and on what evidence?","id":"evidence-neighbours-q01","kind":"provenance"}]}]}]}]},"agentConduct":{"may":["Compute and explain Eulerian numbers and their identities, showing working.","Distinguish them clearly from Euler numbers and other similarly named sequences."],"mustNot":["Confuse Eulerian numbers with Euler numbers or the number e.","Present unverified identities as proven.","Use inconsistent indexing without stating the convention."],"requiresHuman":["Publishing new results under someone else's name."]},"ethics":{"considerations":["Correct attribution and naming matter in mathematics, where similar names cause frequent errors."],"affectedParties":["Students","Mathematicians"]},"owners":{"steward":"Nobody owns them; they are held in common as mathematics.","roles":[],"masterSystems":[]},"relations":[{"target":"Q16317911","type":"parent","note":"registry parent class"}],"interaction":{"identity":{"applicability":"required","items":["Vercy registry: vr.tr.eulerian-number","Wikidata: Q1373849 (https://www.wikidata.org/wiki/Q1373849)"]},"properties":{"applicability":"not-applicable","items":["Permutation size: n: nonnegative integer; count of distinct elements","Statistic index: k: integer interpreted under the recorded convention","Counting convention: descents; ascents; excedances; runs with explicit translation; other explicitly defined statistic","Eulerian value: Dimensionless nonnegative integer for an exact count","Family interpretation: ordinary; explicitly defined generalization; unresolved","Verification status: unevaluated; computed; independently checked; conflicting","Defining reference: Read source with definition, notation and formula location"]},"recognition":{"applicability":"not-applicable","items":[]},"capabilities":{"applicability":"required","items":["Resolve notation and translate a documented indexing convention.","Compute an exact indexed value or a finite row.","Check a supplied table against boundary values and Eulerian identities.","Derive descent probabilities after establishing uniform sampling of permutations.","Extract or assemble Eulerian polynomial coefficients under an explicit exponent convention.","Flag ambiguous family assignments or incompatible source values."]},"hazards":{"applicability":"required","items":["Errors from mixing indexing conventions.","Confusion with similarly named numbers."]},"interfaces":{"applicability":"not-applicable","items":[]},"context":{"applicability":"required","items":["Eulerian numbers occur in enumerative combinatorics when counting permutations by their numbers of ascents or descents.","They are used in permutation probability calculations and polynomial identities, with the indexing convention stated before values are compared."]}},"sources":[{"title":"Wikidata item Q1373849: Eulerian number","url":"https://www.wikidata.org/wiki/Q1373849","note":"identity and sense of the item"},{"title":"Wikipedia: Eulerian number","url":"https://en.wikipedia.org/wiki/Eulerian_number","note":"general description of the item"}],"openQuestions":["Does the registry intend only ordinary Eulerian numbers, or also second-order and other generalized families?","Which notation and column convention should become canonical across Vercy's existing mathematical publications?","Does an existing world model already own this concept, requiring a registry link rather than another publication?","Which independently checked tables and executable implementations should supply reference values and computation evidence?"],"resources":{"spec":"/models/things/publications/thing-q1373849/spec.json"},"provenance":{"origin":"thing registry research (pass 2)","builtFrom":["models/things/publications/thing-q1373849/spec.json"],"providers":["Codex"],"researchStatus":"unreviewed","generatedAt":"2026-09-12T09:33:28Z","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":"not-applicable","interaction.capabilities":"filled","interaction.hazards":"filled","interaction.interfaces":"not-applicable","interaction.context":"filled","sources":"filled"},"notes":{"_gate":"Published through the thing publication gate 1.0 on 2026-10-07T10:10:26+00:00: completeness 1.00; sense: Wikidata description; 2 live sources; review by codex: pass","interaction.properties":"Plane XCT: no invented physical properties.","interaction.interfaces":"No standard, law or official classification governs Eulerian numbers as mathematical objects.","interaction.recognition":"An Eulerian number is an abstract mathematical quantity without an intrinsic perceptible appearance."},"score":1.0}}