Kostka number¶
The nonnegative integer K_{λμ} counting semistandard Young tableaux of shape λ and weight μ, also a Schur-to-monomial coefficient.
Core Idea¶
A Kostka number K_{λμ} counts semistandard Young tableaux with a fixed diagram shape λ and fixed label multiplicities μ. Rows may stay equal or increase; columns must increase strictly. The count is a nonnegative integer determined by both constraints.
The same numbers appear as coefficients when Schur symmetric functions are written in the monomial basis. Dominance order controls whether a same-size partition pair can have a positive count; the diagonal case K_{λλ}=1 is a useful check, not the general rule.
Structural Signature¶
Sig role-phrases:
- Partition shape lambda — Fixes the Young diagram and its row and column positions. It is constitutive. Counterfactual: Change the shape and the set of tableaux being counted changes.
- Weight mu — Specifies how many entries of each label appear. It is constitutive. Counterfactual: Without content multiplicities the finite set to count is underdefined.
- Semistandard order — Requires nondecreasing rows and strictly increasing columns. It is constitutive. Counterfactual: If the order rule changes, a different tableau enumeration is counted.
- Enumeration value — Counts valid fillings and can be read as a symmetric-function coefficient. It is output. Counterfactual: A number unrelated to the specified valid tableaux is not this Kostka number.
- Dominance condition — Tests when a same-size partition pair has a positive count. It is diagnostic. Counterfactual: Ignoring shape-weight compatibility can posit a positive count for an impossible filling.
What It Is Not¶
- Not a partition count. It counts valid fillings of one specified shape with one specified weight.
- Not any Young tableau count. Row and column inequalities are essential.
- Not always positive. Dominance and equal total size constrain existence.
- Not generally one. The diagonal identity is a special case.
- Closest near-miss. A filling with the right shape and label counts but equal entries down one column looks close; it fails the strict-column condition and contributes nothing to K_{lambda,mu}.
Scope of Application¶
- Algebraic combinatorics. Enumerates semistandard fillings for specified shape and content.
- Symmetric functions. Reads coefficients in the Schur-to-monomial expansion.
- Representation theory. Interprets the same integers as specified multiplicities under the relevant module construction.
- Sanity checks. Uses dominance and diagonal cases to test an enumeration.
Clarity¶
Write λ for shape and μ for label multiplicities, require equal total boxes and entries, then impose weak rows and strict columns. A raw partition number or a filling that breaks those inequalities is not K_{λμ}. The diagonal count one and dominance positivity are checks under stated partition assumptions, not formulas for every pair.
Manages Complexity¶
Kostka notation replaces a potentially large list of admissible tableaux with one integer tied to precise shape, content, and order rules. The dual coefficient interpretation lets combinatorial and algebraic calculations check one another without losing the conditions that define the counted set.
Abstract Reasoning¶
- Fix a Young diagram of partition shape λ and a weight μ with the same total size.
- List or characterize fillings using exactly the required label multiplicities.
- Reject any filling with a decreasing row or non-increasing column.
- Count the survivors as K_{λμ} or derive the equivalent Schur expansion coefficient.
- Check the diagonal and dominance special cases without substituting them for enumeration.
Knowledge Transfer¶
The exact count travels from tableaux to symmetric functions and representation multiplicities because those mathematical interpretations encode the same λ, μ, and order-constrained integer. A generic count of arrangements or a different tableau statistic is only analogous and must not inherit Kostka identities automatically.
Examples¶
Canonical¶
For shape (3,2) and weight (1,1,2,1), the source counts three admissible semistandard fillings. The shape fixes five boxes, the weight supplies five entries, and the weak-row/strict-column rules decide which arrangements contribute.
Mapped back: Partition shape lambda → two rows of lengths three and two; Weight mu → one 1, one 2, two 3s, one 4; Semistandard order → rows weakly and columns strictly increasing; Enumeration value → K_{(3,2),(1,1,2,1)} = 3; Dominance condition → positive count for this compatible shape-content pair.
Applied / In Practice¶
With shape and weight both equal to the same partition lambda, each row receives its corresponding repeated label in the unique semistandard filling, giving K_{lambda,lambda}=1. This does not imply other weights give one tableau.
Mapped back: Partition shape lambda → chosen partition diagram; Weight mu → the same partition's label multiplicities; Semistandard order → forced row-by-row filling; Enumeration value → one admissible tableau; Dominance condition → equality in the shape-weight order.
Structural Tensions¶
T1 — Shape Constraint versus Weight Composition. The diagram provides positions while the weight fixes available entries; neither alone determines how many semistandard fillings exist.
Diagnostic: Do the box count and label multiplicities match before enumeration?
T2 — Direct Tableau Count versus Algebraic Coefficient. The same integer can be obtained by explicit fillings or by Schur expansion; the equivalence is a theorem, not permission to count arbitrary partitions.
Diagnostic: Which representation is being used to justify the claimed coefficient?
Structural–Framed Character¶
A provisional portable skeleton is counting configurations under structural and multiplicity constraints. K_{λ,μ} counts semistandard Young tableaux of shape λ and weight μ, equivalently a specified Schur-to-monomial coefficient; no exact enumeration parent is verified.
Evaluative weight: Low; the integer records a count, not quality. Human-practice-bound: Low formally, although mathematicians select indexing conventions. Institutional origin: Algebraic combinatorics supplies terminology; the identities follow proof. Vocabulary travels: Equivalent tableau and symmetric-function interpretations preserve the same indices; a different tableau statistic does not. Import versus recognize: Recognize a Kostka number by shape, weight, and tableau rules; importing the name to any constrained count erases its invariant.
Its character: A formal indexed enumeration with a portable counting skeleton and Young-tableau boundary.
Structural Core vs. Domain Accent¶
Skeletal core. Count configurations satisfying a specified shape and multiplicity rule.
Domain-bound accent. Young diagrams, weakly increasing rows, strictly increasing columns, weight μ, and the Schur coefficient equivalence fix K_{λ,μ}.
Why not prime. Counting is general; these indices and tableau constraints make the named number domain-specific.
Instantiates / Related Primes¶
-
Approved root. Stirling numbers count set partitions and the general partition function counts integer partitions; neither is a strict genus of the shape/weight tableau count. No reviewed live node provides the exact broader Kostka-family genus.
-
Related — Young tableaux, Schur functions, dominance order, and representation multiplicities. These are the carrier, alternate expression, feasibility test, and use, not synonyms.
Neighborhood in Abstraction Space¶
Kostka number sits in a moderately populated region (49th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Matrices, Measures & Numeric Structures (30 abstractions)
Nearest neighbors
- Number of groups of a given order — 0.87
- Cyclic Category — 0.86
- Jacobi coordinates — 0.86
- Matrix Multiplication — 0.86
- Distance Matrix — 0.86
Computed from structural-signature embeddings · 2026-10-08
Not to Be Confused With¶
- Standard Young tableaux. Tell: Is μ the all-ones weight special case?
- Partition function. Tell: Are constrained fillings counted, not integer partitions?
- Stirling number. Tell: Are set blocks being counted instead of tableaux?
- Schur coefficient. Tell: Is the expansion the Schur-to-monomial one with the same indices?
References¶
- Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Kostka_number (revision 1363442895).
- Preserved source candidate: https://eudml.org/doc/148506
- Preserved source candidate: http://www.oup.com/uk/catalogue/?ci=9780198504504
- Preserved source candidate: https://archive.today/20121211053838/http://www.oup.com/uk/catalogue/?ci=9780198504504
The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.