LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6.1-sol)

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.

A partition is determined by the multiset of its node contents

Statement

Let $λ, μ ⊢ n$ . If the multiset of contents ${c (x) : x \in [λ]}$ equals the multiset ${c (y) : y \in [μ]}$ , then $λ = μ$ .

Facts & Assumptions

Given: Partitions $λ, μ ⊢ n$ ; for a partition $ν ⊢ n$ we write $[ν]$ for its Young diagram, $ν^{'}$ for its conjugate, and $ν_{j}^{'} = # {i : ν_{i} \geq j}$ for the height of column $j$ (Partitions, English diagrams, and conjugation).

[F1]

A node of $[ν]$ is a pair $(r, c)$ with $r, c \geq 1$ and $c \leq ν_{r}$ ; its content is $c (r, c) = c - r$ ; the rows of $[ν]$ are weakly decreasing (Partitions, English diagrams, and conjugation).

[F2]

The content of a node and the content vector are as defined in The content of a node and the content vector of a standard tableau; in particular the content map is $c (r, c) = c - r$ on nodes.

Proof

technique · direct

1.1F1F2algebra

For a partition $ν ⊢ n$ and an integer $t \geq 0$ put $n_{t} (ν) : = # {x \in [ν] : c (x) = t}$ . A node of content $t$ has the form $(i, i + t)$ with $i \geq 1$ , and it lies in $[ν]$ exactly when $i + t \leq ν_{i}$ , that is $ν_{i} - i \geq t$ ; hence $n_{t} (ν) = # {i \geq 1 : ν_{i} - i \geq t}$ for every $t \geq 0$ , the count being finite and equal to $0$ for $t > n$ .

1.2F1F2algebra

Similarly, for an integer $s \geq 0$ put $n_{- s} (ν) : = # {x \in [ν] : c (x) = - s}$ . A node of content $- s$ is $(j + s, j)$ with $j \geq 1$ ; it lies in $[ν]$ exactly when $j + s \leq ν_{j}^{'}$ , that is $ν_{j}^{'} - j \geq s$ . Hence $n_{- s} (ν) = # {j \geq 1 : ν_{j}^{'} - j \geq s}$ for every $s \geq 0$ .

1.3F1F2algebra

The partition $ν$ is recovered from the pair of strictly decreasing sequences $a_{1} > a_{2} > \dots > a_{d}$ and $b_{1} > b_{2} > \dots > b_{d}$ by the formula $ν_{r} = (a_{r} + 1) \cdot [r \leq d] + # {c : 1 \leq c < r, b_{c} \geq r - c}$ for every row index $r$ . Indeed, for each diagonal node $(i, i) \in [ν]$ with $i \leq d$ let $A_{i} : = {(i, c) : i \leq c \leq ν_{i}}$ be its arm and $L_{i} : = {(r, i) : i \leq r \leq ν_{i}^{'}}$ its leg; arms and legs have sizes $a_{i} + 1$ and $b_{i} + 1$ , and the $d$ hooks $A_{i} \cup L_{i}$ partition $[ν]$ , because a node $(r, c)$ with $c \geq r$ lies in the arm $A_{r}$ and a node $(r, c)$ with $c < r$ lies in the leg $L_{c}$ . Counting row $r$ therefore gives $ν_{r} = ∣ A_{r} ∣ + # {c < r : (r, c) \in L_{c}} = (a_{r} + 1) \cdot [r \leq d] + # {c < r : b_{c} \geq r - c}$ , since $(r, c) \in L_{c}$ means $c \leq r \leq ν_{c}^{'} = c + b_{c}$ .

2.1step 1.1step 1.2F1algebra

For each row index $i$ put $a_{i} : = ν_{i} - i$ , and for each column index $j$ put $b_{j} : = ν_{j}^{'} - j$ . The row lengths are weakly decreasing, so $a_{i + 1} < a_{i}$ for all $i$ ; the column heights are weakly decreasing as well, so $b_{j + 1} < b_{j}$ for all $j$ . Moreover $a_{i} \geq 0$ exactly for the diagonal rows $i$ with $(i, i) \in [ν]$ , and $b_{j} \geq 0$ exactly for the diagonal columns $j$ with $(j, j) \in [ν]$ , so the two multisets $A (ν) : = {a_{i} : a_{i} \geq 0}$ and $B (ν) : = {b_{j} : b_{j} \geq 0}$ have a common cardinality $d (ν)$ , the number of diagonal nodes. By steps 1.1 and 1.2 the numbers $n_{t} (ν)$ , $t \geq 0$ , determine the multiplicity of every value $t \geq 0$ among the $a_{i}$ , namely $n_{t} (ν) - n_{t + 1} (ν)$ , and hence determine the multiset $A (ν)$ together with $d (ν)$ ; likewise the numbers $n_{- s} (ν)$ , $s \geq 0$ , determine $B (ν)$ .

3.1step 2.1step 1.3given∎

Assume now that the multiset of contents of $[λ]$ equals that of $[μ]$ . Then $n_{t} (λ) = n_{t} (μ)$ for every integer $t$ ; by steps 1.1 and 1.2 this forces $A (λ) = A (μ)$ and $B (λ) = B (μ)$ , including the common cardinality $d$ . Writing both multisets as strictly decreasing sequences $a_{1} > \dots > a_{d}$ and $b_{1} > \dots > b_{d}$ , step 1.3 computes the row lengths of $λ$ and $μ$ by the same formula from the same data, so all row lengths agree and $λ = μ$ .

Remarks

Why the diagonal data are Frobenius coordinates. The numbers $a_{1} > \dots > a_{d}$ and $b_{1} > \dots > b_{d}$ are the arm and leg lengths of the diagonal nodes, the Frobenius coordinates of $ν$ ; the formula of step 1.3 is the usual reconstruction of a partition from them. The lemma says that the content multiset, which records the $a_{i}$ and $b_{j}$ through the diagonal counts of steps 1.1 and 1.2, is equivalent to that data.
Sharper statement. The proof shows the two multisets $A (ν)$ and $B (ν)$ separately, not merely their union; both are needed, since the nonnegative and negative contents determine the arms and the legs respectively.

Depends on

Used by

Dependency tree · two levels

4 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

Garsia, Young Seminormal Representation, Murphy Elements and Content Evaluations, UCSD lecture notes (2003), section 3, printed pp. 18-25 (standard reference, not scraped)
Mathas-Soriano, Seminormal Forms and Gram Determinants for Cellular Algebras, J. reine angew. Math. 619 (2008) 141-173; arXiv:math/0604108, section 2, printed pp. 4-8 (standard reference, not scraped)