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.