ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge 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.

The Plancherel measure on partitions of three

Example

For $n = 3$ the partitions are $(3)$ , $(2, 1)$ and $(1^{3})$ , with $f^{(3)} = f^{(1^{3})} = 1$ and $f^{(2, 1)} = 2$ ; hence $P_{3} (3) = P_{3} (1^{3}) = \frac{1}{6}, P_{3} (2, 1) = \frac{4}{6} = \frac{2}{3},$ and $\frac{1}{6} + \frac{4}{6} + \frac{1}{6} = 1$ , recovering The Plancherel weights sum to one in the smallest non-uniform case.

Facts & Assumptions

Given: the Plancherel weights $P_{n} (λ) = (f^{λ})^{2} / n!$ with $f^{λ}$ the number of standard $λ$ -tableaux (The Plancherel measure on the partitions of $n$ , Standard polytabloids form a basis of a complex Specht module).

[F1]

The partitions of $3$ are $(3)$ , $(2, 1)$ and $(1^{3})$ , where $(1^{3})$ is the column $(1, 1, 1)$ (Partitions, English diagrams, and conjugation).

[F2]

For every $n \geq 0$ and $λ ⊢ n$ , $f^{λ} = n! / \prod_{x \in [λ]} h (x)$ with the empty product equal to $1$ ; in particular $f^{(3)} = 3! / (3 \cdot 2 \cdot 1) = 1$ and $f^{(1^{3})} = 3! / (3 \cdot 2 \cdot 1) = 1$ (The hook length formula).

[F3]

The shape $(2, 1)$ has hook lengths $3$ in the top-left box, $1$ in the top-right box and $1$ in the bottom box, so $f^{(2, 1)} = 3! / (3 \cdot 1 \cdot 1) = 2$ ; explicitly its standard tableaux are the ones with first row $(1, 2)$ and second row $(3)$ , and with first row $(1, 3)$ and second row $(2)$ (The hook length formula, Standard polytabloids form a basis of a complex Specht module).

[F4]

The Plancherel weights of any order sum to one (The Plancherel weights sum to one).

Verification

technique · direct

1.1givenF1F2F3

The three shapes: by [F1] the partitions of $3$ are exactly $(3)$ , $(2, 1)$ and $(1^{3})$ , and [F2] and [F3] give $f^{(3)} = 1$ , $f^{(2, 1)} = 2$ and $f^{(1^{3})} = 1$ ; in particular each $f^{λ}$ is a positive integer and the standard-tableau counts are as displayed.

2.1givenstep 1.1algebra

The weights: by definition of the Plancherel measure $P_{3} (λ) = (f^{λ})^{2} / 3!$ with $3! = 6$ , step 1.1 gives $P_{3} (3) = 1 / 6$ , $P_{3} (2, 1) = 4 / 6 = 2 / 3$ and $P_{3} (1^{3}) = 1 / 6$ ; summing, $1 / 6 + 4 / 6 + 1 / 6 = 6 / 6 = 1$ .

3.1givenF4step 2.1algebra∎

Conclusion: the computed weights are the values displayed in the Example, their sum is one as predicted by [F4], and the middle shape carries four times the weight of either extreme shape, so $n = 3$ is the smallest case exhibiting non-uniformity of the Plancherel weights.

Depends on

Used by

The RSK shapes of the six permutations of S₃ Example

Dependency tree · two levels

28 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

Dan Romik, The Surprising Mathematics of Longest Increasing Subsequences, Cambridge University Press 2015; author-hosted manuscript of 20 August 2014 (363 pp.) (standard reference, not scraped)
Vladimir Ivanov and Grigori Olshanski, Kerov's central limit theorem for the Plancherel measure on Young diagrams, arXiv:math/0304010; survey-paper version in Symmetric Functions 2001, NATO Science Series II 74 (2002), 93-151 (standard reference, not scraped)