TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Link states are counted by a ballot-number difference

Statement

Let $n \geq 0$ and let $t$ satisfy $0 \leq t \leq n$ and $n - t$ even. Put $a = (n - t) / 2$ , the number of arcs in a link state, and use the convention $(\binom{n}{a - 1}) = 0$ when $a = 0$ . Then the link-state basis $T_{n, t}$ of $W_{n, t} (R, δ)$ has cardinality

$∣ T_{n, t} ∣ = (\binom{n}{a}) - (\binom{n}{a - 1}) .$

In particular, $W_{n, t}$ is free of this rank over $R$ , independently of $δ$ .

Facts & Assumptions

Given: $n \geq 0$ , an allowed defect number $t$ , $a = (n - t) / 2$ , and the link-state basis $T_{n, t}$ .

[F1]

A link state is a bottom matching with $t$ defects and $a = (n - t) / 2$ arcs whose defects extend to the ordered top endpoints by noncrossing propagating strands; equivalently, no defect lies inside an arc (Temperley–Lieb link-state standard modules by defect number).

[F2]

The binomial coefficient $(\binom{n}{k})$ is the cardinality of the set of $k$ -element subsets of ${1, \dots, n}$ (The set $[A]^{k}$ of $k$ -element subsets and the binomial coefficient $(\binom{n}{k}) : = ∣ [n]^{k} ∣$ ).

Proof

technique · direct

1.1F1givenconstruct

Encode each arc by an opening symbol at its left endpoint and a closing symbol at its right endpoint; encode each defect by an opening symbol. This gives a word of length $n$ with $a + t$ openings and $a$ closings. Every prefix has at least as many openings as closings, because each closing site in that prefix is paired with a distinct earlier opening site in the same prefix.

2.1step 1.1construct

Conversely, scan any word with $a + t$ openings, $a$ closings and nonnegative prefix heights from left to right. Pair each closing symbol with the most recent unmatched opening. The prefix condition guarantees such an opening exists at every closing, and the final number of unmatched openings is $t$ . The resulting arcs are nested or disjoint, so they form a noncrossing matching; the unmatched openings are its defects. This reverses step 1.1, giving a bijection between $T_{n, t}$ and those words.

3.1F2step 2.1algebra∎

There are $(\binom{n}{a})$ words with $a$ closing positions, by [F2]. For $a \geq 1$ , reflect openings and closings through the first prefix whose height is $- 1$ . This is a bijection from words that violate the prefix condition to all words with $a - 1$ closings: the reflected word ends at height $t + 2 \geq 2$ and therefore has a first prefix of height $1$ , whose reflection is the inverse. When $a = 0$ there are no violating words, agreeing with the stated convention. Subtracting the bad count $(\binom{n}{a - 1})$ from the total gives the displayed formula.

Remarks

The literature often indexes the same link states by p, the number of arcs, and gives dim W_{n,t}=binom(n,p)−binom(n,p−1). Here p=a and t=n−2p, so the defect and arc conventions agree exactly.

Depends on

Used by

Dependency tree · two levels

14 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

David Ridout and Yvan Saint-Aubin, Standard Modules, Induction and the Structure of the Temperley-Lieb Algebra (standard reference, not scraped)
John Enyang and Frederick M. Goodman, Cellular bases for algebras with a Jones basic construction (standard reference, not scraped)