ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passaudited 2026-09-27

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.

Small Kostka numbers

Example

Write a filling of a diagram with at most two rows by its rows, so that $11 / 2$ denotes the tableau whose first row is $1, 1$ and whose second row is $2$ . For $n = 3$ the small Kostka numbers are $K_{(2, 1), (2, 1)} = 1, K_{(2, 1), (1, 1, 1)} = 2, K_{(3), (2, 1)} = 1, K_{(1, 1, 1), (2, 1)} = 0.$ The witnesses are: the single filling $11 / 2$ for shape $(2, 1)$ and content $(2, 1)$ ; the two fillings $12 / 3$ and $13 / 2$ for shape $(2, 1)$ and content $(1, 1, 1)$ ; and the single filling $112$ for shape $(3)$ and content $(2, 1)$ . There is no semistandard filling of the column $(1, 1, 1)$ with content $(2, 1)$ .

Facts & Assumptions

Given: The partitions $(2, 1)$ , $(3)$ , $(1, 1, 1)$ and $(2, 1)$ of $3$ , and the filling notation of the Example section.

[F1]

A semistandard tableau of shape $λ$ and content $μ$ is a filling of $[λ]$ such that the entry $i$ occurs exactly $μ_{i}$ times, entries weakly increase along each row and strictly increase down each column; $K_{λ, μ}$ is the number of such fillings (Semistandard tableaux and Kostka numbers).

[F2]

A filling of content $(1^{n})$ is semistandard exactly when it is standard, so $K_{λ, (1^{n})} = f^{λ}$ (Semistandard tableaux and Kostka numbers).

Verification

technique · direct

1.1

For shape $(2, 1)$ and content $(2, 1)$ the filling carries two $1$ 's and one $2$ , so the single entry $2$ occupies one of the three boxes; placing it in the first box of the top row gives $21 / 1$ , whose top row violates weak increase since $2 > 1$ ; placing it in the second box of the top row gives $12 / 1$ , whose column has entries $1, 1$ and violates strict increase; placing it in the bottom box gives $11 / 2$ , whose rows are weakly increasing and whose column entries $1 < 2$ strictly increase, and no other filling is available, so $K_{(2, 1), (2, 1)} = 1$ , realized by $11 / 2$ .

givenF1

1.2

For shape $(2, 1)$ and content $(1, 1, 1)$ the fillings are the bijections of the three boxes onto ${1, 2, 3}$ , six in all; each of $12 / 3$ and $13 / 2$ is semistandard, since its rows are weakly increasing and its column entries are $1 < 3$ and $1 < 2$ , while $21 / 3$ , $31 / 2$ and $32 / 1$ have a top row that is not weakly increasing and $23 / 1$ has column entries $2, 1$ , which are not strictly increasing, and hence $K_{(2, 1), (1, 1, 1)} = 2$ , in agreement with $K_{λ, (1^{n})} = f^{λ}$ and $f^{(2, 1)} = 2$ .

F1F2

1.3

For shape $(3)$ the diagram is a single row, whose weak increase forces the entries to be sorted, so the filling of content $(2, 1)$ must be $1, 1, 2$ , that is $112$ , and this filling is semistandard because a one-row diagram has no column condition; hence $K_{(3), (2, 1)} = 1$ .

givenF1

1.4

For shape $(1, 1, 1)$ the diagram is a single column of three boxes, and strict increase down the column forces the three entries to be pairwise distinct; the content $(2, 1)$ supplies only the two distinct labels $1$ and $2$ , with $1$ repeated twice, so no filling of that content is semistandard and $K_{(1, 1, 1), (2, 1)} = 0$ .

givenF1

2.1

Steps 1.1, 1.2, 1.3 and 1.4 compute the four displayed numbers $1, 2, 1, 0$ , by checking all three content fillings in step 1.1 and all six in step 1.2, and by using the row and column conditions in steps 1.3 and 1.4 to leave respectively one and no semistandard fillings. Each of the latter two shapes has three fillings of content $(2, 1)$ before imposing those conditions. ∎

step 1.1step 1.2step 1.3step 1.4F1F2

Depends on

Semistandard tableaux and Kostka numbers

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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 Craven, Groups, Geometries and Representation Theory - Section 2.4, printed pp. 28-29 (PDF pp. 30-31) (standard reference, not scraped)