Triple

T1249035
Position Surface form Disambiguated ID Type / Status
Subject multinomial theorem E26831 entity
Predicate usedIn P98 FINISHED
Object enumerative combinatorics
Enumerative combinatorics is a branch of mathematics focused on counting and characterizing discrete structures, often using generating functions, bijections, and algebraic techniques.
E141901 NE FINISHED

How this triple was built (4 steps)

Every LLM step that produced this triple, in pipeline order — named-entity classification, the disambiguation choices (the exact options shown, with the pick highlighted), and the generated description. The batch + timestamp of each is in the Provenance table below.

NER Named-entity recognition gpt-5-mini
Instruction
Given a phrase, classify it is english named entity (e.g., persons, organizations, works of art) in Latin script, or not (e.g., literals, dates, URLs, verbose phrases). For disambiguation, the statement where the phrase occurs as object is also given. Please return a JSON object with `phrase` (string, the phrase being analyzed) and `is_ne` (boolean, indicating whether the phrase is a Named Entity).
Input
Phrase: enumerative combinatorics | Statement: [multinomial theorem, usedIn, enumerative combinatorics]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: enumerative combinatorics
Context triple: [multinomial theorem, usedIn, enumerative combinatorics]
  • A. Concrete Mathematics
    Concrete Mathematics is a widely respected textbook by Ronald Graham, Donald Knuth, and Oren Patashnik that blends continuous and discrete mathematics with an emphasis on problem-solving and rigorous analysis, especially for computer science applications.
  • B. Rogers–Ramanujan-type identities
    Rogers–Ramanujan-type identities are a class of deep q-series and partition identities generalizing the classical Rogers–Ramanujan identities, with rich connections to combinatorics, number theory, and modular forms.
  • C. Pascal's identity
    Pascal's identity is a fundamental combinatorial formula that relates adjacent binomial coefficients and underlies many proofs and properties of binomial expansions.
  • D. multinomial theorem
    The multinomial theorem is a fundamental algebraic formula that generalizes the binomial theorem to express powers of sums with any number of terms using multinomial coefficients.
  • E. Bernoulli numbers
    Bernoulli numbers are a sequence of rational numbers that play a central role in number theory and analysis, especially in formulas for sums of powers of integers and in the study of special functions like the Riemann zeta function.
  • F. None of above. chosen
  • G. Unsure - the case is ambiguous/there is not enough information to decide.
NEDg Description generation gpt-5.1
Instruction
Generate a one-sentence description of the target entity. 
You are given a context triple in the form (subject, predicate, object), where the object is the target entity. 
# Instructions
Use the triple to infer relevant information about the entity. Describe the entity based on what is most defining, well-known. 
Avoid repeating the information from the triple, unless really essential.
# Response Format
Return only the sentence: "Description: [one-sentence description of the target entity]"
Input
Entity: enumerative combinatorics
Triple: [multinomial theorem, usedIn, enumerative combinatorics]
Generated description
Enumerative combinatorics is a branch of mathematics focused on counting and characterizing discrete structures, often using generating functions, bijections, and algebraic techniques.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: enumerative combinatorics
Target entity description: Enumerative combinatorics is a branch of mathematics focused on counting and characterizing discrete structures, often using generating functions, bijections, and algebraic techniques.
  • A. Concrete Mathematics
    Concrete Mathematics is a widely respected textbook by Ronald Graham, Donald Knuth, and Oren Patashnik that blends continuous and discrete mathematics with an emphasis on problem-solving and rigorous analysis, especially for computer science applications.
  • B. Rogers–Ramanujan-type identities
    Rogers–Ramanujan-type identities are a class of deep q-series and partition identities generalizing the classical Rogers–Ramanujan identities, with rich connections to combinatorics, number theory, and modular forms.
  • C. Pascal's identity
    Pascal's identity is a fundamental combinatorial formula that relates adjacent binomial coefficients and underlies many proofs and properties of binomial expansions.
  • D. multinomial theorem
    The multinomial theorem is a fundamental algebraic formula that generalizes the binomial theorem to express powers of sums with any number of terms using multinomial coefficients.
  • E. Bernoulli numbers
    Bernoulli numbers are a sequence of rational numbers that play a central role in number theory and analysis, especially in formulas for sums of powers of integers and in the study of special functions like the Riemann zeta function.
  • F. None of above. chosen

Provenance (5 batches)

The batch behind each pipeline step, in order, with when it ran. Timestamps are batch-level — stages were processed in waves, so the object chain (NER → NED1 → NEDg → NED2) reads in order, but predicate / elicitation batches can sit in a different wave.

Step Stage Batch ID Status When
creating Elicitation batch_69a49487a9c48190ba9b05348fd1b53f completed March 1, 2026, 7:33 p.m.
NER Named-entity recognition batch_69a4bf83b32c81908648e5748b897247 completed March 1, 2026, 10:36 p.m.
NED1 Entity disambiguation (via context triple) batch_69ac8f7f59148190a9f1cbd4af8a115e completed March 7, 2026, 8:50 p.m.
NEDg Description generation batch_69ac8ff7ae0c81908ca8ace1f4159383 completed March 7, 2026, 8:52 p.m.
NED2 Entity disambiguation (via description) batch_69ac905e121c819093c9d960b53a45b7 completed March 7, 2026, 8:53 p.m.
Created at: March 1, 2026, 7:47 p.m.