LemmaStatement: Literature-sourcedProof: Literature-sourcedPipeline-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 recording tableau is standard

Statement

Let $w = (w_{1}, \dots, w_{N})$ be a word of pairwise distinct real numbers and define $P_{0} : = \emptyset$ and $P_{k} : = P_{k - 1} \leftarrow w_{k}$ for $1 \leq k \leq N$ (Row insertion and the bumping route). Let $Q_{k}$ be the filling of $[shape (P_{k})]$ that carries the label $k$ in the box added at step $k$ and the label $j$ in the box added at step $j$ for $j < k$ ; the boxes added at the successive steps are distinct addable nodes by Monotonicity of the bumping route and standardness of the output. Then every $Q_{k}$ is a standard tableau with entries $1, \dots, k$ of the same shape as $P_{k}$ ; in particular $Q_{N}$ is a standard tableau of size $N$ .

Facts & Assumptions

Given: A word $w = (w_{1}, \dots, w_{N})$ of pairwise distinct real numbers and the tableaux $P_{0}, \dots, P_{N}$ , with the box $b_{k}$ added at step $k$ and the fillings $Q_{k}$ .

[L1]

$P_{k}$ is standard and $b_{k}$ is an addable node of $[shape (P_{k - 1})]$ ; consequently $[shape (P_{k})] = [shape (P_{k - 1})] \cup {b_{k}}$ , the node $b_{k}$ is the end of its row and of its column of $[shape (P_{k})]$ , and the shape grows by exactly one box at each step (Monotonicity of the bumping route and standardness of the output).

[L2]

A standard tableau is a bijection from its diagram to an initial segment ${1, \dots, m}$ with strictly increasing rows and columns (Tableaux and standard tableaux).

[L3]

A node $(i, λ_{i} + 1)$ addable for $λ$ satisfies $i = 1$ or $λ_{i - 1} > λ_{i}$ , so after insertion it has no box to its right and, by weak decrease of the rows, no box below it (Removable and addable nodes, Partitions, English diagrams, and conjugation).

Proof

technique · induction

1.1baseL2given

Base: $Q_{0}$ is the empty filling of the empty shape, which is standard with entry set $\emptyset$ , and $shape (Q_{0}) = shape (P_{0})$ .

1.2ihgiven

Induction hypothesis: suppose $Q_{k - 1}$ is a standard tableau with entries $1, \dots, k - 1$ of shape $shape (P_{k - 1})$ .

2.1step 1.2L1L3

The box $b_{k}$ is the end of its row and of its column in $[shape (P_{k})]$ by [L1], so in $Q_{k}$ the new label $k$ has no right and no lower neighbour; its left neighbour and its upper neighbour, if present, carry labels $< k$ , and every comparison not involving $b_{k}$ is one already present in $Q_{k - 1}$ . Hence, with $k$ the largest label, rows and columns of $Q_{k}$ are strictly increasing.

3.1step 1.2step 2.1L1

The filling $Q_{k}$ is a bijection from $[shape (P_{k})]$ onto ${1, \dots, k}$ : $Q_{k - 1}$ is a bijection onto ${1, \dots, k - 1}$ by the induction hypothesis, the shapes differ by the single node $b_{k}$ , and $Q_{k}$ agrees with $Q_{k - 1}$ off $b_{k}$ and carries label $k$ on it.

4.1step 1.1step 1.2step 2.1step 3.1discharge-induction∎

By steps 2.1 and 3.1 and the induction hypothesis, $Q_{k}$ is a standard tableau with entries $1, \dots, k$ of shape $shape (P_{k})$ , for every $k \leq N$ , and induction over $k = 0, \dots, N$ proves the assertion; in particular $Q_{N}$ is standard of size $N$ .

Depends on

Used by

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Sources

C. Schensted, Longest Increasing and Decreasing Subsequences, Canadian Journal of Mathematics 13 (1961), 179-191 (13 pp.) (standard reference, not scraped)
Donald E. Knuth, Permutations, Matrices, and Generalized Young Tableaux, Pacific Journal of Mathematics 34 (1970), 709-727 (standard reference, not scraped)
Charlotte Chan, Representation Theory of Symmetric Groups (Oxford Hilary Term 2011 lecture notes, 40 PDF pp.) (standard reference, not scraped)