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.
Scope of Application¶
These mathematical uses preserve the same shape, weight, and semistandard order constraints.
- 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¶
Specify partition shape λ, label multiplicities μ, and equal total box and entry counts. Include fillings with weakly increasing rows and strictly increasing columns; exclude an arbitrary partition count or a filling that breaks either rule. The diagonal value K_{λλ}=1 and dominance positivity apply under their stated conditions, not as a substitute for counting 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.
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