CorollaryStatement: Literature-sourcedProof: Literature-sourcedPipeline-generatedjudge pass (gpt-6.1-sol)

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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.

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The sum of squares of the standard tableau numbers

Statement

For every $n \geq 0$ , $\sum_{λ ⊢ n} (f^{λ})^{2} = n!,$ where $f^{λ}$ is the number of standard $λ$ -tableaux and $f^{\emptyset} = 1$ , $0! = 1$ .

Facts & Assumptions

Given: An integer $n \geq 0$ , the set $X_{n}$ of words $(w_{1}, \dots, w_{n})$ of pairwise distinct real numbers with ${w_{1}, \dots, w_{n}} = {1, \dots, n}$ , and for each partition $λ ⊢ n$ the number $f^{λ}$ of standard $λ$ -tableaux.

[L1]

The Robinson-Schensted map $w \mapsto (P (w), Q (w))$ is a bijection from $X_{n}$ onto the set of pairs $(P, Q)$ of standard tableaux of the same shape $λ ⊢ n$ (The Robinson-Schensted correspondence).

[F1]

A standard $λ$ -tableau is a filling of the Young diagram of $λ$ by $1, \dots, n$ , each once, increasing along rows and columns; $f^{λ}$ is the number of such tableaux, and $f^{\emptyset} = 1$ is the number of fillings of the empty diagram (Tableaux and standard tableaux).

[F2]

A word of $X_{n}$ is determined by the function $i \mapsto w_{i}$ , which is a bijection of ${1, \dots, n}$ ; conversely every such bijection gives a word in $X_{n}$ , and $X_{0}$ consists of the empty word alone (The Robinson-Schensted correspondence).

Proof

technique · direct

1.1F1algebra

The shapes $λ ⊢ n$ are pairwise distinct as subsets of the plane, so the sets of pairs of standard tableaux of shape $λ$ are pairwise disjoint over $λ ⊢ n$ .

1.2F1algebra

For fixed $λ ⊢ n$ the pairs $(P, Q)$ of standard $λ$ -tableaux are exactly the choices of a standard $λ$ -tableau $P$ followed by an independent choice of a standard $λ$ -tableau $Q$ , so there are $f^{λ} \cdot f^{λ} = (f^{λ})^{2}$ of them.

2.1L1F2step 1.1step 1.2algebra

By [L1] the map $w \mapsto (P (w), Q (w))$ is a bijection from $X_{n}$ onto the disjoint union over $λ ⊢ n$ of the sets counted in step 1.2; comparing cardinalities and using that the bijections of ${1, \dots, n}$ are $n!$ -in-number (with $0! = 1$ ) gives $n! = ∣ X_{n} ∣ = \sum_{λ ⊢ n} (f^{λ})^{2}$ .

3.1F1F2given∎

At $n = 0$ the only partition is $\emptyset$ and the sum is the single term $(f^{\emptyset})^{2} = 1^{2} = 1 = 0!$ , so the identity holds at the boundary.

Depends on

Used by

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Sources

David A. Craven, Groups, Geometries and Representation Theory (Spring Term 2013 lecture notes, 42 pp.) (standard reference, not scraped)
Charlotte Chan, Representation Theory of Symmetric Groups (Oxford Hilary Term 2011 lecture notes, 40 PDF pp.) (standard reference, not scraped)
Jeremy L. Martin, Lecture Notes on Algebraic Combinatorics (263 pp.) (standard reference, not scraped)